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GEOMETEICAL CONICS. 
 
By the same Author. 
 AN ELEMENTAEY TREATISE ON CONIC 
 
 SECTIONS. Twelfth Edition. Crown 8vo. 7s. Qd. 
 Key. 10s. 6d 
 
 AN ELEMENTARY TREATISE ON SOLID 
 
 GEOMETRY. Fourth Edition. Crown 8vo. 9s. Qd. 
 
 ELEMENTARY ALGEBRA. Fifth Edition. 
 
 Globe 8vo. 4s. Qd. Key. 10s. M. 
 
 For the Second Edition the whole book was thoroughly 
 revised, and the early chapters remodelled and simplified, 
 and chapters on Logarithms and Scales of Notation were 
 added. The number of examples was also very greatly 
 increased. 
 
 A TREATISE ON ALGEBRA. Fourth Edition. 
 
 Crown 8vo. 7s. 6d Key. 10s. 6d 
 
 This book now contains a Chapter on the Theory of 
 Equations. 
 
 LONDON: MACMILLAN AND CO. 
 
GEOMETKICAL CONICS 
 
 BY 
 
 CHARLES SMITH, M.A. 
 
 MASTEB OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE. 
 
 ^^ at TBM 
 
 fUiriVBRSII' 
 
 Honbon : 
 MACMILLAN AND CO. 
 
 AND NEW YORK. 
 1894 
 
 [All Eights reserved.] 
 
0A^^^ 
 
 r 
 
 Catnfaritige : 
 
 ^ 
 
 PRINTED BY C. J. CLAY, M.A. ANB SONS, 
 AT THE UNIVERSITY PRESS. 
 
PEEFACE. 
 
 XN the first chapter of the following work I have proved 
 -■- some of the properties of conies by methods which are 
 applicable to all the different forms, and have then 
 proceeded to consider the three types of conies in detail 
 in separate chapters. The book may, however, be said to 
 have two beginnings; for the chapter on the Parabola is 
 so written that it may be read first by those who prefer 
 this arrangement. 
 
 In the first chapter I have made use of the Eccentric 
 Circle of Boscovioh to obtain a general proof of the 
 fundamental theorem that the ratio of the rectangles 
 contained by the segments of two chords of a conic drawn 
 in fixed directions is constant. This method was first 
 introduced into English text-books by Dr C. Taylor in his 
 very able and comprehensive Treatise on the Ancient and 
 Modern Geometry of Conies. 
 
 I have discarded the usual method of treating a hyper- 
 bola as if it were two conies, namely, the curve itself and 
 
 5-^0 1-7 
 
VI PREFACE. 
 
 the conjugate hyperbola. This will, I hope, meet with 
 the approval of teachers. 
 
 Numerous easy examples are given under the different 
 propositions throughout the book; the examples at the 
 end of each chapter are more difficult, and should not be 
 attempted on the first reading. 
 
 The important properties of confocal conies are dis- 
 cussed at length, and complete solutions of other interesting 
 and important theorems and constructions are given. 
 
 I have endeavoured to make the book suitable for 
 beginners, but have included chapters on Cross-ratios, 
 Reciprocation and Conical Projection, which are now 
 included in the first year's course at Cambridge. 
 
 CHARLES SMITH. 
 
 October, 1894. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 PAGE 
 
 General Properties of Conics ....:. 1 
 
 CHAPTER II. 
 
 The Parabola .30 
 
 CHAPTER III. 
 The Ellipse 82 
 
 CHAPTER IV. 
 The Hyperbola 122 
 
 CHAPTER V. 
 
 Sections of a Cone 180 
 
 CHAPTER VI. • 
 
 Orthogonal Projection. Similarity of Curves. Curva- 
 ture OF Conics 189 
 
 CHAPTER VII. 
 
 Cross-ratios. Involution. Anharmonic Properties of 
 
 Conics • -204 
 
 CHAPTER VIII. 
 
 Reciprocation. Conical Projection 234 
 
taSIVBHSITW 
 
 CHAPTER I. 
 
 1. Definitions. A Conic Section^ or Conic, is 
 
 the curve traced out by a point which moves in the plane 
 containing a fixed point and a fixed straight line, in such 
 a manner that its distance from the fixed pointis in a 
 constant ratio to its perpendicular distance from the fixed 
 straight line. 
 
 The fixed point is called a focus, the fixed straight 
 line is called a directrix, and the constant ratio is called 
 the eccentricity of the conic. 
 
 It will be shewn hereafter that if a right circular cone 
 be cut by any plane, the section is in all cases a conic as 
 defined above. It was as sections of a cone that the 
 properties of these curves were first investigated. 
 
 A conic is called an ellipse, a parabola, ol- a hyperbola 
 according as its eccentricity is less than, equal to, or greater 
 than unity. 
 
 2. Our object is to investigate the principal geome- 
 trical properties of conies. We first find the position and 
 shape of the different curves. 
 
 S. C. ^ 1 
 
CONICS. 
 
 Prop. I. To find the points in which a conic, whose 
 focus, directrix and eccentricity are given, is cut hy 
 the straight line through its focus 'perpendicular to the 
 directrix. 
 
 Let 8 be the focus and KK the directrix of a conic. 
 
 Draw through >Si the line ZSZ' perpendicular to the 
 directrix and cutting it in the point X. 
 
 In 8X take a point A such that the ratio of SA to 
 AX may be equal to the eccentricity of the conic ; then 
 A will be a point on the conic. 
 
 Divide X8 externally in A' such that 
 SA' :A'X = ^A'.AX] 
 then A' will also be a point on the conic. 
 
 A S 
 
 A' Z 
 
 , Fig.l. 
 
 K' 
 
 A' 
 
 AS 7i 
 
 , Fig. 2. 
 
 The point' ^' will be in X8 produced if SA<AX, 
 that is if the conic is an ellipse (Fig. 1) ; and the point A' 
 will be in SX produced if BA >AX, that is if the conic is 
 a hyperbola (Fig. 2). 
 
 If, in either case, we suppose the eccentricity of the 
 conic to become more and more nearly equal to unity, the 
 distance of the point A^ from the focus will increase with- 
 out limit. Hence one of the points in which the line ZZ' 
 cuts the parabola whose focus is ;Sf and directrix KK' is at 
 an infinite distance from S.. 
 
 Thus the line through the focus of a conic perpendicular 
 to its directrix cuts the conic in two points which are on the 
 same side or on opposite sides of the directrix according as 
 the conic is an ellipse or a hyperbola. Also the line 
 through the focus of a parabola perpendicular to its 
 directrix will only meet the curve in one point at a finite 
 distance from the focus. 
 
CONICS. 
 
 3 
 
 3. Prop. II. To find the points in which a conic, 
 whose focus, directrix and eccentricity are given, is cut hy a 
 straight^Hne parallel to its directrix. 
 
 Let S be the focus and KK^ the directrix of the conic. 
 
 Draw through ^ the line ZSZ' perpendicular to the 
 directrix and cutting it in the point X. 
 
 Through any point iV in Z8Z' draw the line HNH' 
 parallel to the directrix. 
 
 With centre S and radius >SfQ such that the ratio of 
 iSQ to NX is equal to the eccentricity, describe a circle 
 cutting HH' in the points P, P'\ then P, P' will be points 
 on the curve. 
 
 M 
 
 K 
 
 
 I 
 
 I 
 P 
 
 
 
 
 
 N 
 
 V 
 
 
 
 Z X 
 
 A 
 
 ^. 
 
 
 If 
 
 
 A' 
 
 Z' 
 
 M' 
 
 
 
 
 / 
 
 I^ 
 
 / 
 
 
 
 K' 
 
 
 I 
 
 
 For, if PM, P'M' be perpendiculars on the directrix, 
 
 PM=NX = P'W. 
 
 Hence SP : PM= SF : PW = SQ : NX. 
 
 It will be proved in Article 6 that the circle described as 
 above will intersect the line HH' provided that N is between A and 
 A' in the case of an ellipse, and is not between A and A' in the 
 case of a hyperbola. 
 
 Since SP = SP', and X8N is perpendicular to PP', 
 PN must be equal to NP'. 
 
 Now when a straight line is so related to a curve 
 that corresponding to any point of the curve there is 
 another point such that the chord joining the two points 
 
 ^-^-^ 0? THB 
 
 'uyiVBRSITTl 
 
4 CONICS. 
 
 is bisected perpendicularly by that straight line, then the 
 curve is said to be symmetrical about the straight line, 
 and the straight line is called an axis of the curve. 
 
 Thus a conic is symmetrical about the straight 
 line through its focus perpendicular to its directrix, 
 which line is accordingly called an axis of the curve. 
 
 A point where an axis cuts the curve is called a 
 vertex. 
 
 Thus the points A, A' in Art. 2 are vertices of the conic. 
 Sometimes the line AA\ terminated by the vertices, is called the 
 axis. 
 
 4. 
 
 have 
 
 If, in Article 2, G be the middle point oi AA\ we 
 
 SA :AX = SA' :A'X; 
 .'. SA : AX = 8A + SA' : AX + A'X 
 = SA'-SA :A'X-AX. 
 
 K 
 
 A S 
 
 K' 
 
 Z' 
 
 Z' 
 
 Fig. 1. 
 Fig. 2. 
 
 Hence in the ellipse (Fig. 1) 
 
 SA :AX = 2GA : 2CX = 2GS : 2GA ; 
 and in the hyperbola (Fig. 2) 
 
 SA : AX = 2GS y2GA = 2GA : 2GX. 
 Thus in both curves we have 
 
 GS:GA = GA :GX = SA : AX, 
 whence also GA^ = GS . OX, 
 
 and GS:GX = GS' : GA^ 
 
 = SA':AX\ 
 
 >/ 
 
CONICS. 5 
 
 If the eccentricity of the conic be equal to the ratio 
 g : 1, so that SA : AX = e :1, and therefore SA = e . AX, 
 the above relations may be expressed more shortly thus 
 
 G8=e.GA, GA = e.GX, GS.GX = GA\ 
 
 and GS = eKGX. 
 
 Ex. 1. If two points on a conic be equally distant from a focus, the 
 line joining the points must be parallel to the directrix, and the focal 
 distances of the points must be equally inclined to the axis. 
 
 Ex. 2. The directrix of a conic is given and also two points on the 
 curve ; shew that the focus must lie on a fixed circle. 
 
 Ex. 3. Find the focus of a conic whose directrix is given and also 
 three points on the curve. How many conies can be drawn to satisfy the 
 given conditions ? 
 
 Ex. 4. A circle passes through a fixed point and cuts a fixed straight 
 line at a given angle; shew that the centre of the circle must lie on a 
 fixed hyperbola. 
 
 Ex. 5. S is the focus of a conic and P is any point on the curve. 
 Shew that the locus of the middle point of SP is a conic of the same 
 eccentricity whose focus is S and whose corresponding directrix is midway 
 between S and the directrix of the original conic. 
 
 Ex. 6. S is the focus of a conic and P is any point on the curve ; find 
 the locus of the point Q, which divides SP so that the ratio SQ : SP is 
 constant. 
 
 Ex.7. Shew that two conies with the same focus and directrix cannot 
 intersect. 
 
 Ex. 8. Determine the directrix of a conic having given the focus, the 
 eccentricity and two points on the curve. How many positions of the 
 dii'ectrix can there be ? , 
 
 Ex. 9. Find the focus of a conic having given the directrix, the 
 eccentricity and two points on the curve. How many possible positions 
 of the focus can there be ? 
 
 Ex. 10. Shew that the length of a focal chord of a conic is to 
 twice the distance of its middle point from the directrix in the ratios of 
 the eccentricity. 
 
6 
 
 CONICS. 
 
 5. Prop. Ill, To find the points in which a conic, 
 whose focus, directrix and eccentricity are given, is cut hy 
 any line parallel to its axis. 
 
 Let B be the focus and KK' the directrix. 
 
 Find A, A' the vertices of the conic, and let G be the 
 middle point oi AA'. 
 
 Let MM' be a line parallel to the axis cutting the 
 directrix in M. We have to find the points of intersection 
 of MM' and the conic. 
 
 Join MS, and let it meet the lines through A, A' 
 parallel to the directrix in the points a, a' respectively. 
 K 
 
 Then, by similar triangles, 
 
 8a'.aM = BA'.AX) 
 and Sa' : a'M= SA' : A'X = SA:AX', 
 
 /. Sa:aM=Sa' :a'M. 
 
 Hence *, if we describe a circle on aa' as diameter, and 
 Q be any point on this circle, then will 
 
 SQ:QM = Sa:aM=SA : AX. 
 
 If then the line MM' cut the circle in P, P' we shall 
 have 8P:PM = SP' : P'M = SA : AX, 
 
 * Taylor's Euclid, p. 426. 
 
CONICS. 
 
 7 
 
 Hence, as P'PM is perpendicular to the directrix, the 
 points P, P' must he on the conic. 
 
 Now, if be the centre of the circle, will be the 
 middle point of aa , and the line through parallel to the 
 directrix will therefore be equidistant from Aa and A' a, 
 and thus will pass through 0, the middle point of AA'. 
 But 0(7 is perpendicular to PP' and therefore bisects PP\ 
 in F suppose. Thus the middle point of PP' is on the 
 line parallel to the directrix through the fixed point G. 
 
 Hence the conic is symmetrical about the line 
 through G parallel to the directrix, which is therefore 
 also an axis. 
 
 The axes perpendicular and parallel to the directrix 
 are distinguished from one another by being called respec- 
 tively the transverse axis and the conjugate axis. 
 
 6. If, in the preceding Article, the circle cut the lines 
 Aa, A' a again in the points 6, h' respectively, ah' and ha' 
 are both parallel to AA', since the angles ah' a' and aha' 
 are right angles. 
 
 In the case of the ellipse [Fig. 1], PP' is farther from 
 the centre of the circle than ah' or a'h, since a and a are 
 on the same side of M. Hence PP' < ah', that is PP' <AA! , 
 whence it follows that the ellipse lies entirely hetween the 
 lines Aa and A' a. 
 
8 
 
 CONICS. 
 
 If P be any point on the ellipse and PM be the per- 
 pendicular on the directrix; then, 
 since P is between the lines Aa 
 and A'a\ PM must be less than 
 A'X, and therefore SP less than 
 SA\ Thus every point of an el- 
 lipse is at a limited distance from 
 the focus. Hence an ellipse is a 
 closed oval curve. 
 
 If BOB' be parallel to the directrix, and B, B' be such 
 that SB = SB' == e . CX ; then B, B are points on the 
 ellipse, and are the extremities of the conjugate axis. 
 
 In the case of the hyperbola [Fig. 2], PP' is nearer to 
 the centre of the circle than ah' or a'h, since M is between 
 a and a. 
 
 Hence PP' >AA', whence it follows that the hyperbola 
 lies entirely without the lines Aa and A' a'. Since M is 
 within the circle, the line MM' will always cut the circle 
 in real points; moreover it is easily seen that PP' increases 
 indefinitely as XM is increased. 
 
 Thus the hyperbola is a curve with two distinct 
 branches, as in the figure. 
 
 7. Central Conies. Let P be any point on an 
 ellipse or a hyperbola. Draw through P a line parallel 
 to the directrix cutting AA' m N and the curve again 
 in Q. Then PN-= NQ [Prop. II.]. Now draw through Q 
 a straight line parallel io AA' cutting the curve in R and 
 the line through G parallel to the directrix in if; then 
 Qilf = ilfP [Prop. III.]. 
 
CONICS. 
 
 9 
 
 Since PQ = 2PN, and QR = 2QM=2N'G, it follows 
 that PGR is a straight line, and that CR = PC 
 
 
 
 B 
 
 
 
 (" 
 
 
 / 
 
 P 
 
 K 
 
 \ 
 
 S y/ 
 
 c 
 
 M 
 
 K 1 
 
 
 ^^>--^^_^ 
 
 .^---^ 
 
 Q 
 
 B' 
 
 Thus, if P be any point on an ellipse or a hyperbola, 
 and PC be produced to R so that GR = PG\ then the 
 point R will also be on the conic, so that all chords of the 
 conic drawn through the point G are bisected at G. On 
 this account the point G is called the centre of the 
 conic. 
 
 The ellipse and hyperbola are called central conies 
 to distinguish them from the parabola which has no centre, 
 or rather whose centre is at an infinite distance from the 
 focus. 
 
 Note. The parabola can be considered as the limiting 
 form of an ellipse or of a hyperbola. It is a very instruc- 
 tive exercise to deduce from any property of an ellipse 
 or hyperbola the corresponding property of the parabola, 
 when the properties of the two curves are not precisely 
 the same; this should however be deferred until the 
 geometrical properties of the parabola have been con- 
 sidered in detail in the next chapter. 
 
10 
 
 CONICS. 
 
 8. Prop. IV. To shew that a central conic has two 
 foci and two directrices. 
 
 We first prove as in Art. 5 that an ellipse or hyperbola 
 is symmetrical about the line through G parallel to the 
 directrix. From this it follows that if the points 8', X' 
 be taken on the transverse axis such that OS' = SO and 
 CX^ = XG, the point S' will have the same properties 
 with respect to the curve as the point S has; hence 
 S' will also be a focus of the conic and the corresponding 
 directrix will be the line through X' parallel to the 
 original directrix. 
 
 Thus an ellipse or a hyperbola has two foci and 
 two directrices. 
 
 9. Prop. V. To find the points in which any given 
 straight line cuts a conic whose focus, directrix and 
 eccentricity are given. 
 
 Let S be the focus and KXK' the directrix of the 
 conic. 
 
CONICS. 11 
 
 Let MM' be the given straight line cutting the 
 directrix in M. 
 
 In SX take a point A such that SA : AX may be 
 equal to the given eccentricity, and draw AH parallel to 
 MM' cutting the directrix in H. 
 
 Join >Siif and divide it internally and externally at the 
 points a, a' in the ratio 8 A : AH. Then, if we describe a 
 circle on aa' as diameter, and Q be any point on this 
 circle, the ratio of SQ to QM will be equal to the ratio of 
 8a : aM. 
 
 Let the circle cut the line MM' in the points P, P' ; 
 then P, P' will be points on the conic. 
 
 Draw PL, P'L' perpendiculars on the directrix. 
 
 Then, since P is on the circle whose diameter is aa', 
 
 8P : PM=8a : aM = 8A : AH; 
 
 .'. 8P:8A^PM:AH 
 
 But, from the similar triangles LPM and XAH, 
 
 PM:AH = PL:AX. 
 
 Hence 8P:8A=PL:AX, 
 
 whence it follows that P, and similarly P', is a point on 
 the conic. 
 
 It should be noticed that in the case of the ellipse or 
 parabola the points a, a', and therefore also the points 
 P, P' are on the same side of M, that is on the same side 
 of the directrix. In the case of the hyperbola however 
 the points P, P' will or will not be on the same side of the 
 directrix, according as 8 A is less or greater than AH; 
 and, although 8 A > ^X, it does not follow that 8 A > AH. 
 
 It should also be noticed that if the direction of the 
 chord be changed so that AH may become more and more 
 nearly equal to 8A, the distance 8a' will increase without 
 limit, as will also the distance MP'; and that when the 
 direction is such that >Sf^= 8 A, one of the two points in 
 which MM' cuts the conic will be at an infinite distance 
 from the directrix. 
 
12 
 
 CONICS. 
 
 10. Prop. VI. If a straight line cutSa conic, whose 
 focus is S, in the points P, P' and the directrix in the 
 point D ; then will SD be equally inclined to BP and SP'. 
 
 Join SP, SP\ SD, and draw the perpendiculars PM, 
 P'M' on the directrix. Produce PS and P'8 to meet the 
 conic again in p, p' respectively. Then, by definition, 
 
 SP:PM=^BF '.P'M'', 
 :. SP:SP' = PM:P'M'. 
 But, from the similar triangles MPD, M'P'D, we have 
 
 PM'.P'M' = PD:P'D. 
 
 Hence SP : SP' = PD : P'JD, 
 
 whence it follows that DS bisects the angle PSp' provided 
 P, P' are both on the same side of jD, and that PS bisects 
 the angle PSP' provided P, P' are on opposite sides of 
 the directrix, which last condition can, however, only be 
 the case when the conic is a hyperbola. 
 
 Cor. I. A straight line can only cut a conic in two 
 points. 
 
 For, if DPP'P" is a straight line, the points P, P\ P" being on 
 the conic, the line BS would have to make equal angles with /SP, 
 5P' and SP" ; and this is impossible. 
 
CONICS. 13 
 
 Cor. II. If PBp, P'Sp' be any two focal chords of a 
 conic, the lines PP\ pp' will meet on the directrix, as will 
 also the lines Pp' and P'p. 
 
 For, if PP* meet the directrix in D ; then, as we have just proved, 
 DS will bisect the angle PSp' ; and the line joining S to the point 
 where pp' cuts the directrix will also bisect the angle PSp'. Hence 
 FP' and pp' must cut the directrix in the same point. 
 
 Similarly Pp' and P'p will both meet the directrix in a point D' 
 such that D'S bisects the angle psp'. 
 
 The lines DS, D'S will clearly be at right angles to one another. 
 
 Ex. 1. Shew that, if the focus of a conic be given and also two 
 points on the curve, the directrix must pass through one or other of 
 two fixed points. 
 
 Ex. 2. Find the directrix of a conic having given a focus and three 
 points on the curve ; and shew that three at least, of the four possible 
 conies, must be hyperbolas. 
 
 Ex. 3. Find the directrix of a conic having given a focus, the 
 direction of the transverse axis, and two points on the curve. 
 
 Ex. 4. P, P' are the extremities of a focal chord of a conic, and Q is 
 any other point on the curve ; PQ, P'Q cut the corresponding directrix 
 in K, K' respectively. Shew that KK' subtends a right angle at the 
 focus. 
 
 Ex. 5. PSP' is a focal chord, and ^ is a vertex of a conic ; PA, P'A 
 cut the corresponding directrix in K, K' respectively. Shew that 
 
 KX . XK'=XS\ 
 
 X being the foot of the directrix. 
 
 Ex. 6. P is any point on a conic whose focus is S, and ^ is a vertex 
 of the conic. PA cuts the corresponding directrix in K, and KQ is drawn 
 parallel to the transverse axis to meet PS produced in Q. Shew that the 
 locus of O is a parabola. 
 
 Ex. 7. Find the focus of a conic having given the directrix, a vertex 
 and one other point on the curve. 
 
 Ex. 8. Find the directrix of a conic having given the focus, the 
 vertex, and one other point on the curve. 
 
 Ex. 9. Shew that, if P, P' be the two extremities of any central 
 chord of a conic whose focus is S, then SP + SP' will be constant. 
 
 Ex. 10. Shew that, if two conies have a common directrix, their 
 points of intersection must lie on a circle whose centre is on the line 
 joining their foci. 
 
14 
 
 CONICS. 
 
 11. Definitions. Let two neighbouring points P, P' 
 be taken on any curve, and let the point P' move along the 
 curve nearer and nearer to the point P ; then the limiting 
 position of the line PP', when P' moves up to and 
 ultimately coincides with P, is called the tangent to 
 the curve at the point P. Also the line through any 
 point P of a curve perpendicular to the tangent thereat 
 is called the normal to the curve at the point P. 
 
 12. Prop. VII. The portion of a tangent to a conic 
 intercepted between its point of contact and a directrix^ 
 subtends a right angle at the corresponding focus. 
 
 Let the straight line DPP' cut a conic in the points 
 P, P' and a directrix in the point P, the points P, P' 
 being on the same side of the directrix. Then, if 8 be 
 the focus corresponding to that directrix, and PS produced 
 cut the conic again in p, it can be proved as in Art. 10 
 that D8 bisects the angle P'8p. 
 
 Now let P' move up to and ultimately coincide with 
 P, and let PZ be the ultimate position of the line PP\ 
 that is of the tangent at P. Then DS will always make 
 
 equal angles with PS and Sp, and therefore ultimately, 
 when P' has moved up to P, and B to Z, ZS will make 
 equal angles with PS and Sp. Hence each of the 
 angles ZSP, ZSp is a right angle. 
 Thus ZP subtends a right angle at S, 
 
CONICS. 15 
 
 Conversely, if SZ be drawn perpendicular to SP to meet 
 the directrix in Z, then PZ will be the tangent at P. 
 
 13. Prop. VIII. Tangents atthe extremities of a focal 
 chord of a conic intersect in the corresponding directrix. 
 
 Let P8P' be any focal chord of a conic whose focus is 
 S. Draw SZ perpendicular to PSP' 
 meeting the directrix corresponding 
 to the focus S in Z. 
 
 Then, since the angles ZSP, 
 ZSP' are right angles, ZP and ZP' 
 are tangents. 
 
 Thus the tangents at P and P' 
 intersect on the directrix. 
 
 Conversely, if tangents be drawn 
 to a conic from any point on a 
 directrix, the line joining the points of contact will pass 
 through the corresponding focus. 
 
 If PM, P'M' be the perpendiculars on the directrix ; then M, Z, 
 S, P are cyclic. Hence Z SZPf L MZP according as SPfPM ; 
 
 so also iSZF^lWZP' „ 
 
 Hence I PZP'f a right angle „ „ „ , 
 
 [provided, however, that if the conic is a hyperbola the points P, P' 
 are the same side of the directrix]. 
 
 Def. The focal chord of a conic perpendicular to the 
 transverse axis is called the Latus Rectum. 
 
 From the above it is easily seen that the tangents 
 at the extremities of the latus rectum intersect in the 
 point X. 
 
 14. Prop. IX. If from any point Q the perpen- 
 dicular QM be drawn to the directrix of a conic, and S 
 be the corresponding focus ; then will the ratio SQ : QM 
 be greater or less than the eccentricity according as the 
 point is without or within the conic. 
 
 A point Q is without a conic when the line SQ cuts 
 the conic in one and only one point between S and Q. 
 
16 
 
 CONICS. 
 
 Let Q be without the conic, and let SQ cut the conic 
 in P. Draw PN perpendicular 
 to the directrix, and join SM. 
 Then SM will cut PJST at a 
 point K between P and iV. 
 
 Since KP is parallel to QMy 
 
 SQ:QM=SP:PK, 
 
 But SP : PK is greater than 
 SP:PN; 
 
 .-. BQ : QM is greater than SP : PN, 
 that is greater than the eccentricity of the conic. 
 
 If Q and 8 be on opposite sides of the directrix, and if 
 SQ cut the conic in the two points P, P', of which P is 
 between S and Q. Then, if P'N^ be drawn perpendicular 
 to the directrix, SM produced will cut P'l^' in a point K' 
 between P'and N'; whence it follows as above that SQ : QM 
 is greater than SP' : P'N'. 
 
 It can be proved in a similar manner that if Q be 
 within the conic, SQ : QM will be less than the eccen- 
 tricity of the conic. 
 
 15. Prop. X. If from any point T on the tangent at 
 a point P of a conic, TH he drawn perpendicular to the 
 directrix and TK perpendicular to the focal distance SP ; 
 then will the ratio SK : TH he equal to the eccentricity. 
 
 Let the tangent at P meet the directrix in Z. Draw 
 PN perpendicular to the directrix. 
 
 Then ZS is perpendicular 
 to SP, and therefore parallel 
 toTK. 
 
 Hence 
 
 SK:SP = ZT:ZP 
 
 = TH : PK 
 Hence 
 
 SK:TH=SP'PN 
 
 = SA :AX, 
 
CONICS. 
 
 17 
 
 16. Prop. XI. To draw tangents to a conic from 
 an external point. 
 
 Let T be the external point. Draw TH perpendicular 
 to the directrix, and with 8 as centre describe a circle 
 whose radius is to TH as 8 A : AX. 
 
 Since T is without the conic, 8T : TH is greater than 
 8 A : AX. Hence 8T is greater than the radius of the 
 circle. We can therefore draw two real tangents, TK, 
 TK' suppose, to the circle. 
 
 Draw 8Z, 8Z' parallel to TK, TK meeting the 
 directrix in Z, Z' respectively. 
 
 Join ZT, ZT and produce 8K, 8K' to meet ZT, Z'T 
 respectively in Q, (^\ then TQ, TQ^ will be the tangents 
 required. 
 
 Draw QN, QN' perpendicular to the directrix. 
 Then, since SZ is parallel to TK, 
 
 8Q : 8K = ZQ : ZT= QN : TH; 
 .'. 8Q:QN=8K:TH 
 
 = 8A : AX,hy construction. 
 Hence the point Q is on the conic, and therefore as 
 Z8Q is a right angle, ZTQ is the tangent at Q. Similarly 
 Z'TQ' is the tangent at Q\ 
 
 s. c. 2 
 
18 
 
 CONICS. 
 
 17. Prop. XII. To shew that the two tangents drawn 
 to a conic from an external point subtend equal or supple- 
 mentary angles at a focus. 
 
 Let TQ, TQ' be the two tangents, and let TQ, TQ\ 
 produced if necessary, cut the directrix in Z, Z respec- 
 tively. 
 
 Draw TH perpendicular to the directrix, and TK, TK' 
 perpendicular to 8Qy SQ' respectively. 
 
 Then [Prop. XL] SK : TH = SA : AX 
 
 = SK' : TH. 
 
 Hence SK = 8K\ 
 
 whence it follows that T is on the internal bisector of the 
 angle KSK' and therefore on the internal or on the ex- 
 ternal bisector of the angle QSQ'. 
 
 If the points Q, Q' are on the same side of the direc- 
 trix, and if T be also on that side of the directrix ; then 
 it is easily seen that SK and SQ will be in the same 
 direction and so also will SK' and SQ". Hence in this case 
 TS will bisect the angle QSQ\ 
 
 Also, if Q, Q are on the same side of the directrix and 
 T be on the opposite side; then SK and SQ will be in 
 
CONICS. 19 
 
 opposite directions and so also will SK' and BQ'. Hence 
 in this case also TS will bisect the angle QSQ. 
 
 If, however, the points Q, Q are on opposite sides of 
 the directrix, SK and SQ will be in the same or opposite 
 directions according as SK' and 8Q' are in opposite direc- 
 tions or in the same direction. Hence in this case, that 
 is when Q, Q' are on different branches of a hyperbola, T8 
 will bisect the exterior angle Q8Q'. 
 
 Thus, if TQ, TQ he tangents to a conic whose focus is S, 
 TS will bisect the angle QSQ' unless the conic is a hyperbola 
 and Q, Q' are on opposite branches, in which case TS will 
 bisect the exterior angle QSQ\ 
 
 [The student should draw figures to illustrate the different cases.] 
 
 Cor. If the tangents at the points Q, Q' of a conic 
 intersect at T, and the chord QQ' cut a directrix iri D ; 
 then will DT subtend a right angle at the corresponding 
 focus. 
 
 From the above ST bisects the interior or the exterior 
 angle QSQ' according as Q, Q' are on the same or on 
 opposite sides of the directrix. Also, by Prop. VI., SD 
 bisects the exterior or the interior angle QSQ' according 
 as Q and Q' are on the same or on different sides of the 
 directrix. 
 
 Hence in all cases the lines ST and SD are at right 
 angles. 
 
 Ex. 1. Having given a directrix of a conic and the tangent at a given 
 point on the curve, shew that the locus of the focus corresponding to the 
 given directrix is a circle. 
 
 Ex. 2. Having given a focus of a conic, two points on the conic and 
 the tangent at one of those points ; find the corresponding directrix. 
 
 Ex. 3. Having given a directrix of a conic, two points on the curve 
 and the tangent at one of those points ; find the corresponding focus. 
 
 Ex. 4. Construct a conic having given a focus, the eccentricity and 
 the tangent at a given point. 
 
 Ex. 5. Find the focus of a conic having given the directrix, the 
 eccentricity and the tangent at a given point. 
 
 Ex. 6. PN is the perpendicular from any point of a conic on its 
 transverse axis, and JVP produced cuts the tangent at an extremity of the 
 latus-rectum in the point K. Shew that NK=SP. 
 
 2—2 
 
20 
 
 CONICS. 
 
 18. Prop. XIII. The locus of the middle points of 
 a system of parallel chords of a conic is a straight line 
 through the centre of the conic. 
 
 Let QQ be one of the parallel chords, and let V be the 
 middle point of the chord. 
 
 Draw QM, Q'M\ VN perpendicular to the directrix, 
 and QL, QL' perpendicular to NV, 
 
 Then, since V is the middle point of QQ\ V is also the 
 middle point of LL'. 
 
 Draw through the focus B a line perpendicular to QQ' 
 meeting the corresponding directrix in Z, QQ' in F, and 
 i\rF in iT. 
 
 Then 
 SQ^ - SQ^ ==YQ'^-qY^ = (YQ + Q'Y)(YQ- Q'Y) 
 
 = 4>YV.VQ, 
 [since YQ+ Q'Y=2VQ and YQ- Q'Y=2YV] 
 = ^LV.VK, 
 since Q, L, Y, K are on a circle. 
 
CONICS. 21 
 
 Again, if e be the eccentricity of the conic, 
 Sq^ = e\QM'^ = eKLN\ 
 and 8Q"' = e' . Q'M"' = e^ . L'N\ 
 
 Hence SQ' - SQ"" = e' {LN^ - LN^) = 4<e' . LV . VN. 
 Hence LV . VK=e\LV. VF; 
 
 .', VK = e\VK 
 
 But [Art. 4] G8 = e\ GX, where G is the centre 
 
 of the conic. 
 
 Hence ZVG is a straight line. 
 
 But ^ is a fixed point for all chords parallel to QQ\ 
 and therefore the middle points of all parallel chords of a 
 conic are on the fixed line joining the centre to the point 
 where a directrix is met hy a line through the corresponding 
 focus perpendicular to the chords. 
 
 Def. The locus of the middle points of a system of 
 parallel chords of a conic is called a diameter. 
 
 From the above we see that all diameters of a conic are straight 
 lines through the centre of the conic, and therefore in the particular 
 case of a parabola the diameters are straight lines which cut the axis 
 at an infinite distance from the focus, and are thus parallel to the 
 axis. This result can also be deduced by putting e = 1 in the relation 
 VK= e^. VN, for if VK= VN, Z and N must coincide. 
 
 Cor. I. If any diameter of a conic meet the curve 
 in P, then will the tangent at P be parallel to the system of 
 chords bisected by that diameter. 
 
 For if the line through P parallel to the chords 
 bisected by the diameter CF meet the conic again in the 
 point R, the middle point of PR will be on GP, and 
 therefore at P, which can only be the case when the 
 points P and R coincide. 
 
 We know that the diameters AG A', BGB' bisect chords 
 parallel to BB', A A' respectively; it therefore follows that 
 the tangents at A and A' are parallel to BGB\ and the 
 tangents at B and B' are parallel to AG A'. 
 
22 coNics. 
 
 Since a diameter meets a central conic in two points 
 
 it follows that the tangents at those two points are 
 
 parallel to one another. 
 
 This however follows immediately from the fact that the centre is 
 the middle point of all chords of the conic which pass through it. 
 For, if PCP', RGB' be any two chords, PR will be equal and parallel 
 to P'R'. Hence if R move up to P, and therefore R' to P', in the 
 limit the tangent at P will be parallel to the tangent at P'. 
 
 Cor. II. The tangents at the extremities of any chord 
 of a conic meet on the diameter which bisects the chord. 
 
 Let QQ', RR' be any two parallel chords of a conic ; 
 and let V, W be their middle points. Then WV is a 
 diameter of the curve. Let RQ cut If F in T. 
 
 Then WT:VT = RW:QV\ 
 
 :. WT:VT=RW:Q'V. 
 
 Hence, as R'W is parallel to Q'V, R'Q'T is a straight 
 line. 
 
 Thus RQ, R'Q' meet on the diameter which bisects 
 QQ', and this is true for all positions of the parallel chord 
 RR'. 
 
 Now let RR' move up to and ultimately coincide 
 with QQ' ; then the lines RQ, R'Q' will ultimately be the 
 tangents at Q, Q' respectively. 
 
 Hence the tangents at Q, Q' will meet on the diameter 
 which bisects QQ'. 
 
CONICS. 
 
 23 
 
 Ex. 1. Having given an arc of a conic drawn on paper; find its 
 centre by means of a ruler and compasses. 
 
 Ex. 2. Shew that the transverse axis is the greatest and the conjugate 
 axis is the least diameter of an ellipse. 
 
 Ex. 3. Shew that the transverse axis is the longest and the latus- 
 rectum the shortest focal chord of an ellipse. 
 
 19. Prop. XIV. If the normal at any 'point P of a 
 conic cut the transverse axis in G, then will SG : SP be 
 equal to the eccentricity of the conic. 
 
 Let the tangent at P cut the directrix corresponding 
 to S in the point K. 
 
 Draw PM perpendicular to the directrix, and join BM. 
 
 Then, since the angles KBP and KMP are right 
 angles, K, S, P, M are on a circle. 
 
 Hence- ZSMP = ZSKP 
 
 = compt. of Z SPK = Z SPG. 
 
 Also Z SPM^Z. P8G, since PM and SG are parallel. 
 
 The triangles SPM, G8P are therefore similar, and we 
 have 
 
 SG:SP = SP : PM^SA: AX. 
 
24 
 
 CONICS. 
 
 20. Prop. XV. If the normal at any point P of a 
 conic cut the transverse aods in G ; then will the projection 
 of PG on SP be equal to the semi-latus-rectum. 
 
 Draw GL perpendicular to SP, and SF perpendicular 
 to SX meeting the curve in F; then SF is the semi-latus- 
 rectum, and we have to prove that PL = SF. 
 
 We can prove, as in Prop. XIV., that the triangles 
 GSP, SPM are similar. 
 
 Hence PG : SM == SP : PM 
 
 = SA '.AX, 
 But, since Z GPL = Z SMP = z XSM, 
 and z GLP = rt. ^=-^ SXM, 
 
 the triangles GLP and SXM are similar. 
 
 Hence PL : SX=^ PG : SM. 
 
 Therefore PL : SX = SA : AX 
 
 = SF:SX. 
 Hence PL = SF = semi-latus-rectum. 
 
CONICS. 
 
 25 
 
 21. Prop. XVI. The semi-latus-rectum is the har- 
 monic mean between the two segments of any focal chord. 
 
 Let PSP' be any focal chord of a conic, and let PBP' 
 produced, if necessary, cut the corresponding directrix 
 in K. Let BF be the semi-latus-rectum. 
 
 Draw PM, P'M' perpendicular to the directrix, and 
 PNy P'W perpendicular to the transverse axis. 
 
 M 
 
 
 
 V 
 
 AIN' 
 
 F X 
 
 
 
 /' 
 
 K 
 
 \ 
 
 Then, since the triangles PBN, P'BN' are similar, 
 8P : 8P' = m : SJSf' 
 
 ^PM-SX:SX-P'M\ 
 But SP = e. PM, SF=e.SX and SP' = e, P'M' ; 
 .-. SP-SF=e(PM-SX), 
 and SF -SF = e {SX - P'M') ; 
 
 .-. PM-SX :SX-P'M' = SP-SF:SF-Sr; 
 
 .'. SP:SP' = SP-8F:SF-SP', 
 
 Thus SP, SF and SP' are in Harmonic Progression. 
 
26 
 
 THE ECCENTRIC CIRCLE. 
 
 THE ECCENTRIC CIRCLE. 
 
 22. Definition. The circle described with any 
 point in the plane of a conic as centre and the ratio 
 of whose radius to the perpendicular distance of the 
 point from the directrix is equal to the eccentricity of the 
 conic, is called the eccentric circle of the point. 
 
 23. By means of the eccentric circle we can find the 
 points in which any given straight line cuts a conic 
 whose focus, directrix and eccentricity are given. 
 
 For, let the line cut the directrix in M. Draw the 
 eccentric circle of any point on the line, and let SM 
 cut the circle in p, p'. Join Op, Op\ and draw SP, SP' 
 parallel to Op, Op' respectively, meeting the line MO 
 in the points P, P' respectively. Then P, P' will be 
 points on the conic. 
 
 Draw PN, P'N' and OK perpendicular to the direc- 
 
 trix. 
 
 Then, since BP is parallel to Op and PN to OK, 
 SP:Op = PM: OM 
 =:^PN:OK; 
 .-. 8P:PN=0p : OK. 
 
THE ECCENTRIC CIRCLE. 
 
 27 
 
 But, by the definition of the eccentric circle, Op : OK 
 is equal to the eccentricity of the conic. Hence P, and 
 similarly P', is a point on the conic. 
 
 24. Prop. XVII. The ratio of the rectangles con- 
 tained hy the segments of two intersecting chords of a conic, 
 which are parallel to two fixed straight lines respectively, is 
 independent of the position of the point of intersection of 
 the chords. 
 
 Let one of the chords through the point cut the 
 conic in P, P' and the directrix in M. Let the line 
 ^M, joining the focus to the point M, cut the eccentric 
 circle of in the points p, p': Then, as in Art. 23, 
 Op, Op' will be parallel to SP, SP' respectively. 
 
 Hence POiOM = Sp :pM, 
 
 and P'O'.OM =Sp':p'M; 
 
 .-. PO . P'O : OM^ = Sp . Sp : Mp . Mp. 
 
 Hence PO . P'O : Sp . Sp' = OM^ : MT% where MT 
 is the tangent from M to the eccentric circle of 0. 
 
 Now, if the line OPP' be fixed in direction, the ratio 
 OM: OK will be constant. Hence, as OK : OT is equal to 
 the eccentricity of the conic, OM : OT will be constant, 
 and therefore, as OTM is a right angle, OMiMT will 
 be constant provided that the chord is parallel to a fixed 
 straight line. 
 
28 THE ECCENTKIC CIRCLE. 
 
 Let another chord through cut the conic in Q, Q and 
 the directrix in M', Let M'T be the tangent from M' to 
 the eccentric circle of and let SM' cut that circle in 
 q, ((. Then, as before, 
 
 qO . qO : Sq . Sq' = Oilf ^ : M'T\ 
 
 But, as before, the ratio OM'' : M'T^ will be constant 
 if the chord OQQ' be parallel to a second fixed straight 
 line. 
 
 Also Sp.8p' = 8q.Sq', 
 
 whence it follows that the ratio PO.P'OiQO. Q'O will be 
 constant for all positions of 0, provided that each of the 
 chords is parallel to a fixed straight line. 
 
 EXAMPLES. 
 
 1. Given the focus and directrix of a conic, shew that the chord of 
 contact of the tangents drawn to the conic from a given point will pass 
 through a fixed point. 
 
 2. PSI" is any focal chord of a conic, and Q is any other point on 
 the curve ; PQ, P'Q cut the directrix corresponding to the focus S in 7f, 
 K' respectively. Shew that the rectangle KX . XK' is constant. 
 
 3. The tangent at any point P of a conic whose focus is S meets the 
 transverse axis in T. Shew that ST is greater than, equal to, or less 
 than SP according as the conic is an elHpse, parabola, or hyperbola. 
 
 4. PSQ is a focal chord of a conic and the tangent at P is cut in T 
 by the line through Q perpendicular to PQ. Shew that TQ is bisected 
 by the directrix. 
 
 5. The focal chord PSP' of a conic, produced if necessary, cuts the 
 directrix in K. Shew that SK is a harmonic mean between PS and P'K. 
 
 6. A chord QQ' of a conic cuts the directrix in Z and the line join- 
 ing the points of contact of the tangents from Z to the conic in K. 
 Shew that ZQ^ ZK, ZQ' are in harmonic progression. 
 
 7. Having given the foci of a central conic and the line joining the 
 points of contact of the tangents drawn to the conic from a given point ; 
 find the directrices. 
 
 8. Draw, parallel to a given straight line, a tangent to a conic whose 
 focus, directrix and eccentricity are given. 
 
EXAMPLES. 29 
 
 9. The normal at the point P on a conic whose focus is S meets the 
 transverse axis in G, and P3I is the perpendicular on the directrix 
 corresponding to the focus S. Shew that, for all positions of P on the 
 curve, MS and PG intersect on a fixed straight line. 
 
 10. Shew that the tangent at a point of a conic is cut by the latus- 
 rectum produced and the directrix in points whose distances from the 
 corresponding focus are in the ratio of the eccentricity. 
 
 11. Shew that the tangents at the ends of any focal chord of a conic 
 cut the latus-rectum produced in points which are equidistant from the 
 focus. 
 
 12. QQ' is any chord of a conic which subtends a given angle at the 
 focus ; shew that the locus of the point of intersection of the tangents at 
 Q and Q' is a second conic ; and that the line QQ' touches a third conic, 
 and also that the three conies have a common focus and directrix. 
 
 13. Two sides of a triangle are given in position, and the third 
 side subtends a constant angle at a fixed point. Shew that the third 
 side always touches a conic of which the fixed point is a focus. 
 
 14. On any tangent to a conic two points K, K' are taken such that 
 KK' subtends a right angle at a focus; shew that the other tangents 
 from iT, K' intersect on the corresponding directrix. 
 
 15. The tangent at a point P of a conic whose focus is 8 cuts the 
 directrix in K and the transverse axis in T, and PM is the perpendicular 
 from P on the directrix. Shew that SM touches the circle SKT. 
 
 16. Shew that, if ^fF be the perpendicular on the tangent at any 
 point P of a conic whose focus is S, and if PM be the perpendicular on 
 the corresponding directrix, the triangles SYX and SPM will be similar. 
 
 17. The tangents at the extremities of a focal chord PSP' of a conic 
 
 > 
 meet in T, and P, P' are on opposite sides of ^Sf ; shew ST^ = PS . SP\ 
 
 according as the conic is an elHpse, parabola or a hyperbola. 
 
 18. PSP' is a focal chord of a conic, and the normals at P, P' meet 
 in ; shew that the line through parallel to the transverse axis wiU 
 bisect PP\ 
 
 19. PSP' is a focal chord of a conic, and the normals at P, P' meet 
 in O, and from the line OK is drawn perpendicular to PSP'. Shew 
 that SP=P'K and SP'=PK. 
 
 20. The normals at any two points P, P' of a conic cut the trans- 
 verse axis in G, G' respectively. Shew that the projections of PG and 
 P'G' on PP* are equal to one another. 
 
CHAPTER II. 
 
 The Parabola. 
 
 25. Definitions. A parabola is the locus of a point 
 which moves in the plane containing a given point and a 
 given straight line, in such a manner that its distance 
 from the given point is equal to its perpendicular distance 
 from the given straight line. The given point is called 
 the focus and the given straight line is called the 
 directrix of the parabola. 
 
 Prop. I. Having given the focus and directrix of a 
 parabola, to find any number of points on the curve. 
 
 Let 8 be the focus and MM' the directrix of a parabola. 
 
 Draw through 8 the line X8N perpendicular to the 
 directrix, meeting the directrix in X. Bisect X8 in A ; 
 then, since 8 A = -4X, J. is a point on the curve. 
 
 Through any point N on XSN draw LNL' perpendi- 
 cular to X8N. 
 
 With centre 8 and radius equal to XN describe a circle 
 to cut LNL' in the points P, P'. 
 
 Draw PM, P'M' perpendicular to the directrix. 
 
 Then, since PM, NX, P'M' are all perpendicular to 
 MXM' and PNP', we have 
 
 8P = XN = MP, and 8P' = XN = M'P'. 
 
 Hence P and P' are points on the curve. 
 
 The necessary and sufficient condition that the circle 
 centre 8 and radius XN should cut the line NL is that 
 
THE PARABOLA. 
 
 31 
 
 XN" should be greater than SX, and this will be the case 
 provided X be taken anywhere to the right of A in the 
 
 M 
 
 L 
 
 P 
 
 
 ^ 
 
 
 X 
 
 M' 
 
 \ 
 
 A S 
 
 N. I 
 
 
 L' 
 
 P' 
 
 figure. Thus any straight line parallel to the directrix of 
 the parabola, and on the same side of the directrix as the 
 focus, will cut the curve in two points provided the dis- 
 tance of the line from the directrix be not less than half 
 the distance of the focus from the directrix. 
 
 Hence a parabola lies entirely on the same side of the 
 directrix as the focus, and extends to an unlimited distance. 
 
 Since SP = SP\ and XSX is perpendicular to PF, 
 PX must be equal to XP\ Now when a straight line is 
 so related to a curve that corresponding to any point of 
 the curve there is another point such that the chord join- 
 ing the two points is bisected perpendicularly by the 
 straight line, then the curve is said to be symmetrical 
 about the straight line, and the straight line is called an 
 axis of the curve. 
 
 We have thus proved that a parabola is symmetrical 
 about the straight line through its focus perpendicidar to its 
 directrix, which line is accordingly called the axis of the 
 curve. 
 
 A point where an axis cuts the curve is called a 
 vertex. 
 
 Thus, in the figure, the point A is the vertex of the parabola. 
 
32 THE PARABOLA. 
 
 26. We have shewn how to determine any number of points on the 
 curve. The curve may be described continuously in the following, 
 manner. 
 
 
 n 
 
 
 
 
 M 
 
 IK 
 
 '\ 
 
 A 
 
 
 
 s 
 
 
 Let a straight rod MK be made to slide with one end M on the 
 directrix and so that the rod is always perpendicular to the directrix. 
 Then, if a string whose length is equal to that of the rod have one end 
 fastened at the extremity of the rod and the other at the focus, and if 
 the string be kept constantly stretched by a pencil in contact with the 
 rod, the point of the pencil will describe a parabola with the given focus 
 and directrix. For, since SP + PK=KM, SP=PM. 
 
 27. It is easily seen that SP is less than PM for any point P within 
 a parabola, and that SP is greater than PM for all points outside. 
 
 For, if P be within the curve, and PM be drawn perpendicular to the 
 directrix it will cut the curve at some point Q between P and M. Then 
 SQ = Q3I, hence PM=SQ+QP, and SQ + QP is greater than SP. 
 
 Similarly for a point outside the curve. 
 
 Ex. 1. If the focal distances of two points on a parabola be equal to 
 one another, the line joining the points must be parallel to the directrix, 
 and the focal distances of the points must be equally inclined to the 
 axis. 
 
 Ex. 2. If the directrix of a parabola be given and also one point on 
 the curve, the focus must lie on a fixed circle. 
 
 Ex. 3. Find the focus of a parabola when the directrix and two 
 points on the curve are given. How many parabolas can be drawn to 
 satisfy the given conditions ? 
 
 Ex. 4. If the focus of a parabola be given and one point on the 
 curve, the directrix will touch a fixed circle. 
 
 Ex. 5. Find the directrix of a parabola when the focus and two 
 points on the curve are given. How many solutions will there be? 
 
 Ex. 6. Find the directrix of a parabola having given the focus, the 
 direction of the axis, and one point on the cmve. 
 
THE PARABOLA. 
 
 33 
 
 Ex. 7. The locus of the centre of a circle which passes through a 
 given point and touches a given straight line, is a parabola. 
 
 Ex. 8. The locus of the centre of a circle which touches a given 
 straight line and a given circle is a parabola whose focus is at the centre 
 of the given circle, and whose directrix is parallel to the given straight 
 line and at a distance from it equal to the radius of the given circle. 
 
 28. Definitions. The perpendicular from any point 
 of a parabola on the axis is called the ordinate of the 
 point. The length of the axis, measured from the vertex, 
 cut off by the ordinate of any point, is called the abscissa. 
 
 Thus, in the figure Art. 25, PN is the ordinate of the point P, and 
 AN is the abscissa. 
 
 A chord PP' perpendicular to the axis is sometimes called a 
 double ordinate. 
 
 Any chord through the focus is called s, focal chord. 
 The focal chord perpendicular to the axis is called the 
 latus-rectum. 
 
 latus-rectum of a 
 
 29. Prop. II. The length of the 
 parabola is four times the distance 
 of the focus from the vertex. 
 
 Let B be the focus, MXM' the 
 directrix, XAS the axis and LSU 
 the latus-rectum. Draw LM, L'M' 
 perpendicular to the directrix. 
 
 Then, by definition, 
 
 SL = LM=SX, 
 and SL' = L'M'^8X. 
 Hence XX' = 2^X = 4^^. 
 
 It should be noticed that two para- 
 bolas which have equal latera-recta are 
 
 equal in all respects. For, since the latera-recta are equal, the 
 distances of the foci from the respective directrices must likewise 
 be equal. One curve may therefore be superimposed on the other 
 (as in Euclid I. 4) so that the directrices are coincident and also the 
 foci ; and in that case the two parabolas will altogether coincide. 
 
 S. C. 
 
34 
 
 THE PARABOLA. 
 
 30. Prop. III. The ordinate at any point of a para- 
 bola is a mean proportional to the abscissa and the latus- 
 rectum. [PN'' = 4}AS . AK] 
 
 Join SP, and draw PM perpendicular to the directrix, 
 and P-AT perpendicular to the axis. 
 
 Then, since SP" = PM' = XK' ; 
 
 and SP'^PN' + SJ^'-; 
 
 .-. PF' = XN'-8N' 
 
 = 4<AS.AX, 
 
 since XN + ^N=2AF and XN - SJSf = 2 AS. 
 
 From the relation PX"" = 4!AS . AN it follows that the 
 square of the perpendicular from any point of a parabola 
 on the axis varies as the perpendicular on a line through 
 the vertex parallel to the directrix. 
 
 Conversely. If a point move in a plane in such a 
 manner that the square of the perpendicular from it on one 
 of two fixed straight lines at right angles to each other varies 
 as the p)erpendicular on the other, the point will describe a 
 parabola. 
 
 Ex. If FN is the ordinate of any point P of a parabola, the 
 locus of the middle point of PN is a parabola. 
 
 For, if Q be the middle point of NP, NQ'' = iNP'- = AS . AN. 
 Hence the locus of Q is a parabola whose vertex is 1, whose axis is 
 ASN, and whose latus-rectum is equal to AS. 
 
THE PARABOLA. 35 
 
 Ex. 1. When the ordinate and the abscissa of a point on a parabola 
 are equal to one another, each is equal to the latus-rectum. 
 
 Ex. 2. Find the abscissa corresponding to a double ordinate whose 
 length is three times the latus-rectum. 
 
 Ex. 3. FNP' is a double ordinate of a parabola whose vertex is A, 
 and the circle APP' cuts the axis in Q. Shew that NQ, is equal to the 
 latus-rectum. 
 
 Ex. 4. PM is the perpendicular from any point P of a parabola 
 upon its directrix, and on PM is taken a point Q such that PQ is 
 constant. Shew that the locus of Q is an equal parabola. 
 
 Ex. 5. The locus of a point which divides the ordinate of a parabola 
 in a constant ratio is another parabola with the same vertex and axis. 
 
 Ex. 6. A is the vertex of a parabola and P is any point on the 
 curve. Shew that the locus of the middle point of AP is another 
 parabola. 
 
 Ex. 7. If S be the focus of a parabola and P be any point on the 
 curve, the locus of the middle point of SP is a parabola whose focus is 
 S and whose directrix is midway between S and the directrix of the 
 given parabola. 
 
 Ex. 8. S is the focus of a parabola and P is any point on the curve ; 
 Q is the point on SP such that SQ : SP is equal to a given ratio. Shew 
 that the locus of Q is a parabola. 
 
 Ex. 9. PM is the perpendicular from any point P of a parabola on 
 its directrix, and 3IP is produced to Q so that MP=PQ. Shew that, if 
 N, N' be the feet of the ordinates through P, Q respectively, SN'=2AN, 
 and that the locus of Q is a parabola whose vertex is S. , 
 
 Ex. 10. PM is the perpendicular upon the directrix of a parabola 
 from any point P on the curve. Shew that the locus of Q, the middle 
 point of MP, is a parabola whose vertex is midway between X and A . 
 
 31. Definitions. Let two points P, P' be taken on 
 any curve, and let the point P' move along the curve 
 nearer and nearer to the point P; then the limiting 
 position of the line PP'y when P' moves up to and 
 ultimately coincides with P, is called the tangent to the 
 curve at the point P. Also the line through any point P 
 of a curve perpendicular to the tangent thereat is called 
 the normal to the curve at the point P. 
 
 3—2 
 
36 
 
 THE PARABOLA. 
 
 32. Prop. IV. (i) If a straight line cut the directrix 
 of a parabola, whose focus is S, in K and the curve in P, P' ; 
 then will KS bisect the exterior angle between SP and SP'. 
 
 (ii) The portion of any tangent to a parabola, inter- 
 cepted between its point of contact and the directrix, sub- 
 tends a right angle at the focus. 
 
 Join SP, SP' and produce PS, P'S to p, p. Draw 
 PM, P'M' perpendicular to the directrix. 
 
 Then, from the similar triangles KPM, KP'M\ we 
 have 
 
 KP:KP' = PM'.P'M' 
 
 Hence 
 
 = >SfP:/SfP',bydef. 
 KS bisects the angle PSp' or P'Sp......(i). 
 
 Now let P' move up to and ultimately coincide with 
 P, and let PZ be the ultimate position of the line PP\ 
 that is of the tangent at P. Then KS will always make 
 equal angles with P'S and Sp, and therefore ultimately 
 
THE PARABOLA. 
 
 37 
 
 ZS will make equal angles with PS and Sp. Hence each 
 of the angles ZSF, ZSp is a right angle. 
 
 Thus ZP subtends a right angle at S (ii). 
 
 Conversely, if 8Z be drawn perpendicular to SP to 
 meet the directrix in Z, then PZ will be the tangent at P. 
 
 33. Prop. V. Tangents at the extremities of a focal 
 chord of a parabola will intersect at right angles in the 
 directrix. 
 
 Let PSP' be any focal 
 chord. Draw SZ perpen- 
 dicular to PSP' meeting 
 the directrix in Z. 
 
 Then, since the angles 
 ZSP, ZSP' are right angles, 
 ZP and ZP' are tangents. 
 
 [Converse of Prop. ly.] 
 
 Thus the tangents at the extremities of the focal chord 
 intersect in. the directrix. 
 
 Draw PM, P'M' perpendicular to the directrix. 
 
 Then, since ZSP and ZMP are right angles, the points 
 Z, S, P, M are on a circle ; and, since SP, PM are equal 
 chords of this circle, the subtended angles SZP and MZP 
 are equal. Therefore ZP bisects the angle MZS. 
 
 Similarly ZP' bisects the angle M'ZS. 
 
 Hence ZP and ZP' are at right angles. 
 
88 THE PARABOLA. 
 
 Ex. 1. PM is the perpendicular on the directrix of a parabola, whose 
 focus is S and vertex A ; shew that MS bisects the angle ASP. 
 
 Ex. 2. If PA cut the directrix in K, KS bisects the exterior angle 
 ASP. 
 
 Ex. 3. P is any point on a parabola whose vertex is A, PM is the 
 perpendicular on the directrix, and PA cuts the directrix in K. Shew 
 that MK subtends a right angle at the focus. 
 
 Ex. 4. PSP' is any focal chord of a parabola, and Q any point on 
 the curve. PQ, P'Q cut the directrix in K, K' respectively. Shew that 
 KK' subtends a right angle at the focus. 
 
 Ex. 5. PSP' is a focal chord of a parabola whose vertex is A, and 
 PA meets the directrix in M ; shew that P'M is parallel to the axis. 
 
 Ex, 6. If two parabolas have a common directrix the line joining 
 their common points will bisect at right angles the line joining their foci. 
 
 Ex. 7. If three parabolas have a common directrix the three common 
 chords of the parabolas taken in pairs will meet in the centre of the 
 circum-circle of the triangle formed by their foci. 
 
 Ex. 8. The tangents at the extremities of the latus-rectum of a 
 parabola pass through the foot of the directrix. 
 
 Ex. 9. A series of parabolas have a common directrix and axis ; 
 shew that they all touch two fixed straight lines at right angles to one 
 another. 
 
 Ex. 10. Any number of parabolas have a common directrix and touch 
 a given straight line ; shew that they all touch another straight line 
 perpendicular to the former, and that their foci lie on a fixed straight 
 line. 
 
 Ex. 11. MKM' is the directrix of a parabola whose focus is S. Shew 
 that the lines bisecting the angles MKS, M'KS will touch the parabola, 
 K being any point on the directrix. 
 
 Ex. 12. S, S' are the foci of two parabolas which have a common 
 directrix; shew that their common tangents meet at right angles at the 
 point of intersection of SS' and the common directrix. 
 
 Ex. 13. Shew that the circle described on any focal chord of a parabola 
 as diameter will touch the directrix. 
 
 Ex. 14. Having given the directrix of a parabola and the tangent at a 
 given point ; find the focus. 
 
 Ex. 15. Having given the directrix and two tangents to a parabola ; 
 find the focus. When are the conditions insufficient ? 
 
THE PAKABOLA. 
 
 39 
 
 34. Prop. VI. The tangent at any point P of a para- 
 bola bisects the angle between SP and the perpendicular 
 PM on the directrix. 
 
 Let the tangent cut the directrix in Z. Join 8Z, and 
 draw PM perpendicular to the directrix. 
 
 Then, since the angles ZSP, ZMP are right angles, the 
 points Z, 8, P, M are on a circle ; and, since >SP, PM are 
 equal chords of this circle, 
 
 ZSZP=:ZMZP. 
 
 Hence the complements of these angles are equal, that is 
 ZSPZ=ZMPZ. 
 Hence PZ bisects the angle SPM. 
 
 Cor. 1. The tangent at A bisects the angle between 
 SA and AX. Hence the tangent at the ver-tex is perpendi- 
 cular to the axis. 
 
 Cor. 2. If ZP be produced to Z\ the angles SPZ' 
 and MPZ' will be equal. 
 
40 
 
 THE PARABOLA. 
 
 35. Prop. VII. If the tangent at any point P of a 
 parabola meet the axis in T, and PN he the ordinate of P ; 
 then will 
 
 SP = ST and TA = AK 
 
 Join SPy and draw PM perpendicular to the directrix. 
 Then Z SPT = Z MPT ; [Prop. VI. 
 
 .-. ZSPT=ZPTS; . 
 
 [since PM and WT are parallel. 
 .-. TS = 8P. 
 And, since T8 = SP = PM = X]Sr- 
 
 /. TA+AS = XA+A]Sr; 
 .-. TA^AJSr. 
 
 Def. The portion of the axis intercepted by the 
 ordinate of any point and the corresponding tangent is 
 called the sub-tangent. 
 
 Hence, in a parabola, the suhtangent is equal to twice 
 the abscissa. 
 
THE PARABOLA. 
 
 41 
 
 36. Prop. VIII. If the normal at P meet the axis 
 in G, and PN be the ordinate of P, then will SG = SP and 
 
 NG = 2AS. 
 
 Join SP, and draw PM perpendicular to the directrix. 
 
 Then Z SPT = Z MPT [Prop. Y I. 
 
 = ZPTS. 
 
 And, since TPG is a right angle and Z 8TP = Z SPT, 
 the complements of these angles are equal, namely 
 
 zSGP^zSPG; 
 
 .'. 8P = SG. 
 Then, since 8G = SP = XF, 
 
 SN+NG = XS + S]Sr; 
 ... NG = XS=2AS. 
 
 Def. The portion of the axis intercepted by the 
 ordinate of any point and the corresponding normal is 
 called the sub-normal. 
 
 Hence at any point of a parabola the subnormal is equal 
 to half the latus-rectum. 
 
42 
 
 THE PARABOLA. 
 
 37. Prop. IX. The locus of the foot of the perpen- 
 dicular from the focus on a tangent to a parabola is the 
 tangent at the vertex, and the length of the perpendicular is 
 a mean proportional between the focal distances of the point 
 of contact and the vertex. 
 
 Join 8P. Draw PM perpendicular to the directrix, 
 and join 8M cutting the tangent at P in Y. 
 
 Then, since BF = PM, and the tangent at P bisects 
 the angle SPM, the tangent must be perpendicular to l^M 
 and will bisect SM in Y. 
 
 Then, since SY=YM and SA = AX, ^F must be 
 parallel to XM. 
 
 Thus Y is on the tangent at the vertex. 
 
 Again, since SYT and YA8 are right angles, the 
 triangles ASY, YST are similar ; 
 
 .-. AS:SY=SY:ST=SY:SP. 
 
 Hence 8Y' = AS.SP. 
 
 The converse of this proposition is very important, 
 namely that if a straight line move in such a manner that 
 the foot of the perpendicular upon it from some fixed point 
 always lies on a fixed straight line; then the moving line 
 will always touch the parabola whose focus is the fixed 
 point and of which the fixed straight line is the tangent at 
 the vertex. 
 
THE PARABOLA. 
 
 43 
 
 Ex. FN, PM are the perpendiculars from any point P of a 
 parabola on the axis and the tangent at the vertex of the parabola ; 
 shew that NM touches a fixed parabola. 
 
 Draw MS' perpendicular to NM meeting the axis in 8'. 
 Then, since NMS' and MAN are right angles, 
 
 But 
 
 MA^=:S'A.AN. 
 MA^=PN^=^S.AN. 
 
 Hence S'A = 4:AS, so that S' is a. fixed point. 
 
 Hence MN touches the parabola whose focus is S' and of which 
 AM is the tangent at the vertex. 
 
 EXAMPLES. VII. AND Vin. 
 
 1. Shew that TSPM is a. rhombus. 
 
 2. Shew that TP and SM bisect each other at right angles. 
 
 3. Shew that MPGS is a parallelogram. 
 
 4. If SPG is an equilateral triangle each side is equal to the latus- 
 rectum. 
 
 5. Shew that the triangles TMS and SPG are equal in all respects. 
 
 6. Shew that the triangles TXM and SNP are equal in all respects. 
 
 7. Shew that the triangles MXS and PNG are equal in all respects. 
 
 8. If GL be perpendicular to SP ; then will PL = NG = 2AS. 
 
 9. Shew that the tangents and normals at the ends of the latus- 
 rectum are along the sides of a square. 
 
 If XH parallel to SP meet PM in H ; then NH will be parallel 
 
 10. 
 
 to PG. 
 
 11. Shew that the locus of the middle point of PG is a parabola 
 whose vertex is S. 
 
44 THE PARABOLA. 
 
 12. Any number of parabolas have the same focus and axis, and 
 tangents are drawn to them from the same point on the common axis ; 
 shew that the points of contact of the tangents he on a circle. 
 
 13. The line joining any point P of a parabola to the vertex cuts the 
 directrix in K. Shew that KS is parallel to the tangent at P. 
 
 EXAMPLES. IX. 
 
 1. Any tangent to a parabola meets the directrix and the latus-rectum 
 produced in points equidistant from the focus. 
 
 2. If the focus S of a parabola be joined to any point M of the 
 directrix, the lines which bisect SM at right angles will touch the 
 curve. 
 
 3. Tangents are drawn to a series of concentric circles at the points 
 where they are cut by a fixed straight line ; shew that these tangents all 
 touch a parabola. 
 
 4. Having given two tangents to a parabola and the focus, find the 
 directrix, 
 
 5. Having given the focus of a parabola and two tangents, find the 
 points of contact of the tangents. 
 
 6. Having given the focus of a parabola and the tangent at a given 
 point, find the directrix. 
 
 7. If two parabolas have a common focus, their common chord will 
 bisect the angle between their directrices. 
 
 8. If two parabolas have a common focus, the line joining it to the 
 point of intersection of the directrices will be perpendicular to their 
 common tangent. 
 
 9. If two equal parabolas have a common focus, their common chord 
 will pass through the focus and will be perpendicular to their common 
 tangent. 
 
 10. is a fixed point, P is any point on a fixed straight line, and 
 on the Hne a point Q is taken such that PQ = OP. Shew that the line 
 through P and the middle point ot OQ touches a parabola whose focus 
 is O. 
 
 11. PN is the ordinate of any point P of a parabola, and a point 
 M is taken on the axis such that AN=N3I. Shew that J/P touches 
 a fixed parabola. 
 
 12. A line is drawn through the focus of a parabola so as to meet 
 the tangent at a point P at a given angle ; shew that the locus of the 
 point of intersection, for different positions of P, is a straight line. 
 
 [The intersection is on the tangent which makes the given angle 
 with the axis.] 
 
THE PARABOLA. 
 
 45 
 
 38. Prop. X. To draw tangents to a parabola from 
 any external point. 
 
 Let T be the external point. Join TS, and upon T8 
 as diameter describe a circle cutting the tangent at the 
 vertex in F, Y'. Then the angles TTS, TYS are right 
 angles, and hence TY, TY' when produced will touch the 
 parabola, for the feet of the perpendiculars from the focus 
 on these lines are on the tangent at the vertex. 
 
 A method of drawing tangents which is applicable to 
 all conies is given in Art. 16. 
 
 Ex. 1. If P be any point on a parabola whose focus is S, the circle 
 whose diameter is SP will touch the tangent at the vertex of the 
 parabola. 
 
 Ex. 2. If T be any point without a parabola, the circle whose 
 diameter is ST will cut the tangent at the vertex of the parabola in two 
 real points. 
 
 Ex. 3. Two real tangents can be drawn to a parabola from any 
 external point. 
 
46 
 
 THE PARABOLA. 
 
 39. Prop. XI. If TP, TF he any two tangents to a 
 parabola whose focus is S, the triangles TPS, T8P' will he 
 similar and SP . SP' = ST'^ ; also the tangents ivill subtend 
 equal angles at the focus. 
 
 Let the tangent at the vertex of the parabola cut the 
 tangents in Y, Y'. Then BY, SY' are perpendicular to 
 TP, TP' respectively. Hence the points ^, F, T, Y' are 
 on a circle. 
 
 Therefore Z STY' = Z SYY'. 
 
 But, if PT cut the axis in t, 
 
 zSYA= z8tY= Z8PT. 
 Hence zSTP'=zSPT. 
 
 Similarly Z 8TP = Z SP'T. 
 
 Hence the remaining angles of the triangles TPS, 
 T8P' are also equal, namely 
 
 zP8T= zT8P'; 
 thus TP, TP' subtend equal angles at the focus. 
 
 Again, since the triangles TP8, T8P' are equiangular, 
 they are similar, and 
 
 8P:8T=8T:8P'; 
 .'. SP.SP'^ST', 
 
THE PARABOLA. 47 
 
 Cor. I. The angle tTP'is supplementary to the sum 
 of BTP' and STP, that is to the sum of SPT and STP, 
 
 Hence Z tTP' = Z TSP = Z r>SF. 
 
 Thus the exterior angle between two tangents to a 
 parabola is equal to the angle either tangent subtends at 
 the focus. 
 
 Cor. II. Since the triangles PST, TSP' are similar, 
 
 Also PS:8T=ST:SP'; 
 
 .-. PS':ST' = PS:SP\ 
 
 Hence TP' : TP'^ = PB : SP\ 
 
 Thus, the ratio of the squares on any two tangents to a 
 parabola is equal to the ratio of the focal distances of their 
 points of contact. 
 
 EXAMPLES. 
 
 1. If TP, TP' be tangents to a parabola, and PP' meet the directrix 
 in Z ; then will SZ be perpendicular to ST. 
 
 [For SZ bisects the exterior angle PSF, and ST bisects the 
 angle PSP'.^ 
 
 2. If TP, TP' be tangents to a parabola whose focus is S, the circles 
 TPS, TP'S wUl touch TP', TP respectively. 
 
 3. Having given two tangents to a parabola and the point of contact 
 of one of them ; shew that the locus of the focus is a circle. 
 
 4. Having given two points on a parabola and the tangents thereat, 
 construct the curve [that is, find the focus and directrix]. 
 
 5. From any point on the axis of a parabola two tangents are 
 drawn ; shew that these tangents cut any other tangent in points equi- 
 distant from the focus. 
 
 6. A variable tangent to a parabola cuts two fixed tangents in the 
 points T, T' ; shew that ST : ST' is constant, S being the focus. 
 
 7. Having given three tangents to a parabola, and the point of 
 contact of one of them; construct the curve. 
 
 8. Having given three tangents to a parabola, and the direction of 
 the axis; construct the curve. 
 
48 
 
 THE PARABOLA. 
 
 40. Prop. XII. If the sides of a triangle touch a 
 parabola^ the circum-circle of the triangle will pass through 
 the focus of the parabola. 
 
 We know that the feet of the perpendiculars from 
 the focus of a parabola lie on a straight line, namely on 
 the tangent at the vertex of the parabola. 
 
 But it is a well-known geometrical theorem that if the 
 feet of the perpendiculars from a point on the sides of a 
 triangle lie on a straight line, that point must be on the 
 circum-circle of the triangle. 
 
 The theorem may also be proved as follows. 
 
 Let PQR be the triangle formed by the tangents, and let the 
 tangent at the vertex of the parabola meet QR, BP, PQ respectively 
 in U, V, W. 
 
 
 Then, since SUQ and SWQ are right angles, S, U, W, Q are 
 cyclic, and therefore 
 
 Z SQW= I SUV. 
 
 And, since SUR and SVR are right angles, S, U, V, R are cyclic, 
 and therefore 
 
 Z SUV= L SRV. 
 
 Hence z SQW= z SRV, i.e. z SQP= z SRP, 
 
 whence it follows that S, Q, P, R are cyclic. 
 
THE PARABOLA. 
 
 49 
 
 41. Prop. XIII. The middle points of a system of 
 parallel chords of a parabola lie on a straight line parallel 
 to the axis of the parabola. 
 
 Let QQ' be any one of the system of parallel chords. 
 Draw QM, Q'M' perpendicular to the directrix. 
 
 Then the circles whose centres are Q, Q' and radii 
 QM, Q'M' respectively will 
 touch the directrix in 
 M, M' respectively and will 
 pass through the focus. 
 
 But it is known that 
 the common chord of any 
 two circles is perpendicular 
 to the line joining their 
 centres and bisects any 
 common tangent. 
 
 Hence the line through 
 S perpendicular to QQ' will 
 bisect MM\ in K suppose. 
 
 Then, since K is the 
 middle point of MM\ the 
 
 line through K parallel to the axis of the parabola, and 
 therefore parallel to MQ and M'Q\ will bisect QQ\ in V 
 suppose. 
 
 The point K is the same for all chords parallel to QQ, 
 whence it follows that the middle points of a system of 
 parallel chords of a parabola are on a straight line parallel 
 to the axis. 
 
 Cor. If the line through V parallel to the axis meet 
 the curve in P, then the tangent at P will be parallel to 
 the system of chords. 
 
 For, if the line through P parallel to the chords whose 
 middle points are on PF, cut the curve again in R ; then 
 the middle point of PR will be on FV^ and therefore at 
 the point P, which can only be the case when the points 
 P, R coincide. 
 
 s. c. 4 
 
50 
 
 THE PARABOLA. 
 
 Def. The locus of the middle points of a system of 
 parallel chords of a parabola is called a diameter. 
 
 From the above we see that all diameters of a para- 
 bola are straight lines parallel to the axis of the parabola, 
 and that the tangent at the point where a diameter meets 
 the curve is parallel to the system of chords bisected by that 
 diameter. 
 
 42. Prop. XIV. The tangents at the extremities of 
 any chord of a parabola meet on the diameter which bisects 
 the chord. 
 
 Let QQ\ RR' be any two 
 parallel chords of a parabola; 
 and let F, W be their middle 
 points. Then VW is a dia- 
 meter of the curve. 
 
 Let QR cut VW in T. 
 
 Then WT:VT=^RW:QV\ 
 
 .', WT'.VT=WR':VQ'. 
 
 Hence TQ'R is a straight line. 
 
 Thus QR, Q'R' meet on the diameter through F; and 
 this is true for all positions of the parallel chord RR'. 
 
 Now let RR move up to and ultimately coincide with 
 QQ'\ then the lines QR, Q'R' will ultimately be the 
 tangents at Q, Q' respectively. 
 
 Hence the tangents at Q, Q' meet on the diameter 
 through F. 
 
THE PARABOLA. 
 
 51 
 
 Def. A line QV, drawn parallel to the tangent at 
 the extremity of any diameter PF of a parabola, is called 
 an ordinate to that diameter. 
 
 The portion of any diameter, measured from its ex- 
 tremity, cut off by any ordinate is called its abscissa. 
 
 43. Prop. XV. If the tangents at the extremities of 
 any chord QQ' of a parabola meet in T, and if the dia- 
 meter through T cut the curve in P and QQ' in V; then 
 willTP = PV. 
 
 We know that the diameter through T will bisect 
 QQ\ and that the tangent at P is parallel to QQ'. 
 
 Let the tangent at P cut TQ in Z, and let the 
 diameter through L cut PQ in W. 
 
 Then LW bisects PQ and is parallel to TPV. 
 Hence TL . LQ = PW :WQ- 
 
 :. TL = LQ. 
 But PL is parallel to QV; 
 
 .-. TP:PV=TL:LQ; 
 ... TP = PV, 
 
 4—2 
 
52 
 
 THE PARABOLA. 
 
 Cor. Since TL =■ LQ, it follows that 
 LQ:LP = LT:LP, 
 
 Hence the ratio of the lengths of two tangents to a 
 parabola is equal to the ratio of the sides of any triangle 
 whose sides are parallel to the tangents and base parallel 
 to the axis of the parabola. 
 
 44. Prop. XVI. The length of the focal chord which is 
 parallel to the tangent at any point P of a parabola is 4>SiP. 
 
 Let Q8Q' be the focal chord parallel to the tangent at 
 P. Then the tangents at Q, Q' intersect at right angles 
 at a point which is on the directrix and also on the 
 diameter through P. 
 
 Hence V is the middle point of the hypothenuse of 
 the right-angled triangle QTQ! ; 
 
 ... QQ'=:20F=2Fr. 
 
 Also ST is perpendicular to QQ\ and P is the middle 
 point of TV. Hence P is the middle point of the hypo- 
 thenuse of the right-angled triangle TB V; 
 .-. TV=2TP=^2P8, 
 .'. QQ' = 2VT = 4>PS. 
 Def. The focal chord parallel to the tangent at P is 
 called the parameter of the diameter through P. 
 
THE PARABOLA. 
 
 53 
 
 45. Prop. XVII. The ordinate to any diameter is 
 a mean proportional to the abscissa and the parameter of 
 that diameter. [QF^ = 45P . PV.^ 
 
 Let the tangent at Q meet the diameter through P in 
 and the axis in T. 
 
 Let the tangents at P and Q meet in K. Join BP, 
 SK, SQ. 
 
 Then PK bisects the angle OPS; 
 
 .-. Z8PK= Z.KPO, 
 Since KP, KQ are two tangents, 
 ZSKP=ZSQK 
 = z STQ 
 = zPOK. 
 Hence the triangles 8PK, KPO are similar 
 .-. 8P'.PK^PK:0P; 
 :. PK' = SP.OP = SP.PV. 
 But, since 0F=20P, QV=2PK, 
 hence QV' = 4<PK' = 4;SfP . P V, 
 
 [Prop. VL] 
 
 [Prop. XL] 
 [Prop. VIL] 
 
54 THE PARABOLA. 
 
 Cor. If QD be drawn perpendicular to the diameter 
 PF of a parabola, and QF be the ordinate to that dia- 
 meter; then will QD'' =^ 4^AS . PV. 
 For, from similar triangles, 
 
 QB^ = Q F^ = PN' : PT' [Fig. Prop. IX.] 
 = SY':BP' 
 = SA:SP. 
 
 From the relation QV^ = 4:SP .PV it follows that if we have any 
 two fixed straight lines OA, OB, and if from a point Q the line QV 
 he drawn parallel to OB to meet OA in V; then, if Q move in such 
 a manner that QV^ varies as OV, the point Q will describe a parabola 
 of which OA is a diameter and OB the tangent at the extremity of 
 that diameter. 
 
 If QV^ varies as OF it at once follows that the perpendicular 
 from Q on OB varies as the square of the perpendicular on OA. 
 Thus a point describes a parabola when it moves in such a manner 
 that its perpendicular distance from one of two given straight lines 
 {inclined at any angle) varies as the square of its perpendicular 
 distance from the other. 
 
 It is also important to notice that, if a straight line cuts two 
 fixed straight lines so that the intercept on one of the fixed lines, 
 measured from their intersection, varies as the square of the intercept 
 on the other; then the line will always touch a fixed parabola of 
 which the fixed lines are respectively a diameter and the tangent at 
 its extremity. 
 
 For, if OP, OK are the fixed straight lines, and the line PK be 
 such that OK^ = OP . OL, where OL is a fixed length ; and, if 
 through a line be drawn making with KO an angle equal to the 
 angle KOP, and on this line a point S be taken such that OS = OL. 
 Then PK will always touch the parabola whose focus is S, which 
 passes through 0, and of which PO is a diameter. 
 
 Ex. 1. If QF be any ordinate to the diameter through a fixed 
 point P on a parabola, and the parallelogram PVQR be completed, 
 the line VR will be a tangent to a fixed parabola. 
 
 For PR^ =QV^= iSP . PV. 
 
 Thus PR^ozPV. Hence RV touches a fixed parabola of which 
 ■VP is a diameter and PR is the tangent at its extremity. 
 
 Ex. 2. OA, OB are two fixed straight lines, and any circle is 
 drawn through the fixed point B to touch OA. Shew that, if the 
 circle touch OA in the point P, and cut OB again in the point Q, 
 the line PQ will always touch a fixed parabola. 
 
THE PARABOLA. 55 
 
 Ex. 3. The algebraic sura of the ordinates, with reference to the axis, 
 of the two extremities of any one of a system of parallel chords of a 
 parabola is constant. 
 
 Ex. 4. If two chords of a parabola make equal angles with the axis, 
 but are not parallel, the algebraic sum of the ordinates of their four 
 extremities will be zero. 
 
 Ex. 5. Draw a focal chord of a parabola of given length. 
 
 Ex. 6. If two focal chords of a parabola be equal they must be 
 equally inclined to the axis. 
 
 Ex. 7. The latus-rectum of a parabola is the shortest focal chord. 
 
 Ex. 8. The line joining the intersection of the tangents to the 
 intersection of the normals at the extremities of any focal chord is 
 parallel to the axis. 
 
 Ex. 9. QF is the ordinate to the diameter through P, and PW is the 
 ordinate to the diameter through Q ; shew that VW is parallel to PQ. 
 
 Ex. 10. TQ, TQ' are tangents to a parabola whose focus is ;Sf. The 
 diameter through T cuts the directrix in /f, and QQ' cuts the axis in O ; 
 shew that SOKT is a parallelogram. 
 
 Ex. 11. OQ, OQ' are two tangents to a parabola, and OM is drawn 
 perpendicular to the axis ; prove that QQ' cuts the axis in a point R such 
 that MR is bisected by the vertex of the parabola. 
 
 Ex. 12. PQ is a chord of a normal at P, and the tangents at P, Q 
 meet in 0. Shew that the diameter through O passes through the other 
 extremity of the focal chord through P. 
 
 Ex. 13. TQ, TQ' are tangents to a parabola whose focus is S, and 
 the diameter through T cuts the curve in P ; shew that 
 
 SQ + SQ'=2TP + 2PS. 
 
 Ex. 14. QV is any ordinate to the fixed diameter PV; shew that 
 the middle point of QF describes a parabola. 
 
 Ex. 15. QQ' is any chord of a parabola parallel to the tangent at a 
 fixed point P, and R is taken on QQ' such that QR : RQ' is constant. 
 Shew that the locus of iJ is a parabola touching the given parabola 
 at P. 
 
 Ex. 16. Given a parabola drawn on paper ; find by means of a ruler 
 and compasses its focus and directrix. 
 
 'T7iri7BRSIT7] 
 
56 
 
 THE PARABOLA. 
 
 46. Prop. XVIII. If two chords of a parabola 
 intersect one another, the rectangles contained hy their 
 segments are in the ratio of the parallel focal chords. 
 
 Let the chords QQ\ qq' intersect in 0. 
 
 Let Fbe the middle point of QQ\ and let PFbe the 
 diameter through V. Let OK be the diameter through 
 0, and draw KM an ordinate to the diameter PV. 
 
 Then QO.OQ'=^QV^-OV^ = QV'-KM'^ 
 
 = 4>SP.PV-4>SP.P3r 
 
 = 4<SP(PV-PM) 
 
 = 4>SP.K0. 
 Similarly, ii pv be the diameter which bisects qq, 
 
 qO.Oq'=4>Sp.KO. 
 
 . .-. QO. OQ' : qO . Oq = 4>SfP : 4% 
 
 But the focal chords parallel to the tangents at P, p 
 are 4^P, 4>Sfp respectively. [Prop. XVI.] 
 
THE PAKABOLA. 67 
 
 If two chords of a parabola be drawn in fixed directions, 
 the parallel focal chords will be the same for all positions 
 of the point of intersection. Thus the ratio of the 
 rectangles contained hy the segments of any two chords 
 of a parabola is independent of the position of their point 
 of intersection, and is therefore equal to the ratio of the 
 squares of the parallel tangents. 
 
 47. Prop. XIX. If a circle cut a parabola in four 
 points, the line joining any two of the points makes the 
 same angle with the axis of the parabola as the line joining 
 the other two points. 
 
 For let K, L, M, iV be the four points of intersection. 
 
 The chords KL and M]}i cannot be parallel unless 
 they are perpendicular to the axis. For the line joining 
 the middle points of parallel chords of a circle is perpen- 
 dicular to the chords ; the chords must therefore by Prop. 
 XIII. be perpendicular to the axis of the parabola. 
 
 Let then KL, MN intersect in 0. 
 
 Then, since the four points are on a parabola, 
 
 OK.OL:OM.ON= 8P : 8P', [Prop. XVIII.] 
 
 where S is the focus and P, P' are the extremities of the 
 diameters which bisect KL, MN respectively. 
 But, since the four points are on a circle, 
 
 OK.OL^OM.ON. 
 
 Hence 8P and 8P' are equal. They must therefore 
 be equally inclined to the axis and on opposite sides of it, 
 and the tangents at P, P' will therefore be inclined at 
 equal angles to the axis, and these tangents are parallel 
 respectively to the chords KL and MN. 
 
 In the same way it can be proved that KM and LN, 
 and also that KN and LM are equally inclined to the 
 axis. 
 
 If the two points K, L coincide, that is if the circle 
 touch the parabola, the tangent at K and the chord MN 
 will make equal angles with the axis. Also the lines KM, 
 KN will make equal angles with the axis. 
 
58 THE PARABOLA. 
 
 Conversely, the four extremities of any two chords of 
 a parabola which are equally inclined to the aosis, but are 
 not parallel, lie on a circle. 
 
 48*. Prop. XX. If any line be drawn through a 
 fixed point to cut a parabola, the tangents at the points of 
 intersection will meet on a fixed straight line. 
 
 Let any straight line through the fixed point cut 
 the parabola in Q, Q'. Let V be the middle point of QQ\ 
 Then the tangents at Q, Q' meet at T on the diameter 
 through F, and TP = PF. 
 
 Let OAO' be the diameter through 0. Let the 
 tangent at A cut TFV in K, and let AN" be parallel to 
 QQ\ Then KP=PK 
 
 Hence 
 
 TK==TP-KP = PV-F]Sr==NV = AO. 
 
 Hence the line through T parallel to the tangent at A 
 will meet the diameter OA in a fixed point 0' such that 
 AO' = KT=OA. 
 
 * The remainder of the chapter should be omitted on the first 
 reading. 
 
THE PARABOLA. ' 59 
 
 Thus for all directions of QQ' the point J' is on a line 
 parallel to the tangent at A, through the point 0' such 
 that ^0' = 0^. 
 
 In the figure the point is within the parabola, but 
 when the point is outside the parabola the above proof 
 applies without change, but in this case two lines can be 
 drawn through each of which will cut the parabola in 
 coincident points, namely the tangents from to the 
 curve; and when the points Q, Q' coincide, T will coincide 
 with them. Thus the locus of T will pass through the 
 points of contact of the two tangents drawn from to 
 the parabola. 
 
 We can prove in a similar manner the converse of the 
 above theorem, namely that if any point he taken on a 
 fixed straight line and tangents he drawn from it to the 
 parabola, the line joining the points of contact of the 
 tangents will pass through a fixed point. 
 
 Def. The straight line which is the locus of the 
 point of intersection of the tangents at the extremities of 
 any chord through a fixed point, is called the polar of the 
 fixed point, and the point is called the pole of the line 
 with respect to the curve. 
 
 From Prop. V. we see that the directrix is the polar of 
 the focus, and that the focus is the pole of the directrix. 
 
 49. Prop. XXI. If any chord of a parabola he 
 drawn through the fi^ed point 0, cutting the curve in Q, Q', 
 the polar of in the point R, and the tangent at A, the 
 extremity of the diameter through 0, in the point T ; then 
 will TO he a mean proportional between TQ and TQ\ and 
 TO will be the harTnonic mean between RQ and RQ\ 
 
 By Prop. XVIII., the ratio of TQ . TQ' to TA' is equal 
 to the ratio of the squares of the parallel tangents. And, 
 by Prop. XV. Cor., the ratio of the tangents parallel to TO 
 and TA is equal to the ratio of TO to TA. 
 
60 
 
 THE PARABOLA. 
 
 Hence 
 
 .: TQ.TQ' = TO' 
 
 .(i). 
 
 Let the diameter through cut the polar of in the 
 point B ; then, since the polar of is parallel to A T, 
 OT:TE=OA : AB. 
 Hence, as OA=AB, OT=TR. 
 But, by (i), TQ:TO = TO : TQ'\ 
 
 .: TQ+TO : TO-TQ=TO + TQ' : TQ - TO. 
 That is, since TO=RT, 
 
 RQ:QO = m'-OQ'\ 
 .-. RQ:RQ' = QO:Oq 
 
 = RO-RQ:RQ'- RO. . .(ii). 
 Hence RQ, RO, RQ' are in harmonic progression. 
 
THE PARABOLA. 
 
 61 
 
 Again, if QN, Q'N' be ordinates to the diameter A 0, 
 and therefore parallel to AT, we have 
 
 AN: AO = TQ:TO, 
 
 and AO'.AN'^TOiTq. 
 
 Hence, from (i), 
 
 AN '. AO = AO : AN\ 
 
 or AN.AN'^^AO'' (iii), 
 
 a result which is often useful. 
 
 50. Propositions VII. and VIII. may be generalised 
 as follows. 
 
 From any point draw ON perpendicular to the aods 
 of a parabola, and OKG perpendicidar to the polar of 
 with respect to the parabola, meeting the polar in K and 
 the axis in G, and let the polar of meet the axis in T. 
 Then will TA = AN ST = SG = SK and NG = 2 AS. 
 
 Let the diameter through cut the curve in P and 
 the polar in V. Let the tangent and normal at P cut 
 the axis in T, G' respectively, and let PN be the ordinate 
 of P. 
 
62 
 
 THE PARABOLA. 
 
 Then T'T=PV=OP = FN\ 
 
 But TA=AN'', 
 
 . rA-rT=AN' -NN'-, 
 :. TA = AN. 
 Again GG' = OP=PV=rT, 
 
 and rS=:SG\ 
 
 .-. rS-rT^SG'-GG'; 
 
 = SK, since I!E"(t is a right angle. 
 Also, the triangles ONG, PN'G' are equal in all re- 
 spects ; 
 
 .-. NG = N'G' = ^AS. 
 
 61. Prop. XXI. Any two tangents to a parabola are 
 cut ^proportionally by any other tangent. 
 
 Let TQ, TQ' be any two tangents to a parabola, and 
 let them be cut by any other tangent in the points B, R' 
 respectively ; then we have to prove that 
 QR'.RT^TE '.RQ. 
 
THE PARABOLA. 63 
 
 Let K be the point of contact of RR'. Draw through 
 R, R', T, K diameters cutting QQ in the points L, L\ V, 
 M respectively. 
 ' Then LM = IQM, and ML' = ^MQ' ; 
 
 ... LL' = \QQ!^QV. 
 
 Hence QL=VL\ 
 
 and therefore also LV=L'Q'. 
 
 Then QR:RT=QL:LV 
 
 = VL' : L'Q' 
 
 = TR : EQ\ 
 We have also 
 
 RK : KE = LM : ML' = QL:LV 
 
 = QR: RT. 
 
 Conversely. If two fixed lines TQ, TQ be cut by a 
 moving line in R, R' respectively so that 
 
 QR:RT=TR' :R'Q\ 
 
 then will the moving line RR' always touch the parabola 
 which touches TQ, TQ' at Q and Q'. 
 
 Ex. 1. Tioo tangents TQ, TQ' to a parabola are cut by any third 
 tangent in R, R' respectively. Shew that, if the parallelogram RTR'O 
 be completed, will be on QQ'. 
 
 Draw RO parallel to TQ' to meet QQ' in 0. Then 
 
 QO : OQ'=QR : RT=TR' : R'Q', 
 
 whence it follows that R'O is parallel to TQ. 
 
 It follows from this that if any parabola be drawn to touch three 
 fixed straight lines, the chord of contact with any two will pass 
 through a fixed point, namely through the other vertex of the paral- 
 lelogram of which the two lines are sides and the third side the 
 diagonal. 
 
64 THE PARABOLA. 
 
 Ex. 2. From any point P on the htise BC of a triangle ABC lines 
 FN, FM are draicn infixed directions to cut AB, AC respectively inN, 
 M. Shew that MN envelops a parabola. 
 
 [When FM, FN are drawn parallel to AB, AC respectively we 
 have the converse of Ex. 1.] 
 
 We first notice that one position of MN will coincide with AB, 
 namely when M is at ^ ; so also one position will coincide with AC. 
 Hence we have to prove that MN touches a parabola which touches 
 AB and AC. This could be at once tested provided the points of 
 contact of AB and AC were known. These points of contact are 
 found by the following constructions. Draw AK parallel to FN 
 meeting BC in K, and KQ parallel to FM meeting ^C in Q. Then 
 draw AK' parallel to FM and K'Q' parallel to FN. 
 
 Now, since Q'K', FN and AK are parallel, 
 
 Q'N '.NA^K'F :FK. 
 And, since K'A, FM and KQ are parallel, 
 
 K'P : FK=AM : MQ. 
 
 Hence Q'N : NA = AM : MQ. 
 
 This shews that MN touches the parabola which touches AQ, 
 AQ' at the points Q, Q'. 
 
 Or thus. Let the circle NFM cut BC in L. Then 
 
 L LNM= L ZyPilf=: constant angle, 
 
 and z LMN= L NFB = constant. 
 
 Hence the triangle LMN is of invariable form. But it is easy to prove 
 that if a triangle LMN of invariable form be inscribed in a given 
 triangle ABC, the circles AMN, BNL, CLM will all meet in a point 
 S, and that this point S is fixed for all positions of LMN. This 
 shews that 3IN touches the parabola whose focus is S and which 
 touches AB, A C. 
 
 The other sides NL, LM also touch fixed parabolas of which S is 
 the focus. 
 
THE PARABOLA. 
 
 65 
 
 CONFOCAL AND COAXIAL PARABOLAS. 
 
 52. Parabolas which have the same focus and their 
 axes in the same straight line have some interesting 
 properties which we proceed to investigate. 
 
 I. Through any point two parabolas will pass having their con- 
 cavities in opposite directions, and the two parabolas will intersect at 
 right angles. 
 
 For, if S be the focus and P any point, and a line be drawn 
 through P parallel to the axis, and points M, M' on different sides 
 of P be taken on this line so that MP=PM'=PS ; it is clear that 
 parabolas with S as focus and lines perpendicular to the axis through 
 M, M' respectively for the directrices will pass through P. The tan- 
 gents at P to the two parabolas will bisect the angles SPM, SPM! 
 respectively and will therefore be at right angles. 
 
 II. One parabola can be drawn to touch any given straight line. 
 Draw SY perpendicular to the given straight line, and produce it 
 
 to M so that SY=YM. Through M draw a line parallel to the axis 
 cutting the given straight line in P. Then it is clear that the para- 
 bola whose focus is S and whose directrix passes through M will touch 
 the given straight line at P. 
 
 III. If a tangent to one parabola of the system be perpendicular 
 to a tangent to another, their point of intersection will be on a straight 
 line midway between their directrices. 
 
 S. C. 
 
66 THE PARABOLA. 
 
 Let QPSP'Q' be a common focal chord of two confocal and coaxial 
 parabolas. The tangents at P, P' intersect at right angles on one 
 directrix, and the tangents at Q, Q' intersect at right angles on the 
 other directrix. 
 
 The tangents at P and Q will be parallel since each tangent makes 
 with the axis an angle equal to half the angle PSN. Similarly the 
 tangents at P' and Q' wiU be parallel. Hence the four tangents are 
 along the sides of a rectangle. If K be the point of intersection of 
 the tangents at Q, P', and KM be drawn perpendicular to QSQ'; 
 then the distance of K from each directrix will be equal to SM. 
 Hence iTis on a straight line midway between the directrices. 
 
 [When the parabolas have their concavities in opposite directions, 
 the perpendicular tangents to the two parabolas have their points of 
 contact on the same side of the axis.] 
 
 IV. If from any point T of a parabola two tangents be drawn to 
 a confocal coaxial parabola^ these tangents will be equally inclined to 
 the tangent at T. 
 
 For if, in the figure to Prop. XI., TL be drawn parallel to the 
 axis; then 
 
 Z FTS= L SPT= L PTL. 
 
 Thus jTP', TP make equal angles with T8 and TL respectively. 
 
 Also the tangent at T to a confocal coaxial parabola through T 
 makes equal angles with TS and TL, whence it follows that TP, TP* 
 make equal angles with the tangent at T to a confocal parabola 
 which passes through T. 
 
 V. If a chord of a parabola vary as the parallel focal chord it 
 will touch a fixed confocal parabola. 
 
 Let QQ' be a chord of a parabola, and 
 PV the diameter which bisects the chord 
 in F. 
 
 Then QV^ = 4:SP.PV. Hence, if QVoc4SP, it follows that PV 
 will vary as SP. 
 
 Let SP cut QQ' in p. Let PM be the perpendicular on the 
 directrix. Join SM, and draw pm parallel to Pilf cutting SM in m. 
 
 Then, since PV and SP make equal angles with the tangent at 
 P, they will make equal angles with QQ'. Hence PV=Pp. 
 
 Therefore, as PF varies as SP, it follows that Sp : SP is constant. 
 Now Sp:pm = SP: PM; .: Sp =pm. And, since Sm : SM is constant, 
 it follows that m is on a fixed straight line parallel to MX. The 
 locus of p is therefore a parabola whose focus is S, and since QQ' 
 bisects the angle between Sp and pm, it follows that QQ' touches this 
 fixed parabola. 
 
THE PARABOLA. 67 
 
 VI. The locus of the 'poles of a given straight line with respect 
 to all parabolas of a confocal system is a straight line perpen- 
 dicular to the given straight line. 
 
 If the given straight line cut the axis in T, and G be taken on the 
 axis such that TS = SG. Then we have seen (in Art. 47) that the 
 pole of the given line with respect to any parabola whose focus is S 
 and axis TSG, is on the line through G perpendicular to the given 
 line. 
 
 VII. The envelope of the polar of a given point with respect 
 to all parabolas of the system is another parabola. 
 
 Let be the fixed point, and let its polar with respect to any 
 parabola whose focus is S cut the axis in T, and let the perpendicular 
 from O on its polar cut the axis in G. 
 
 Then TS=SG. 
 
 Hence, if through T a line be drawn perpendicular to the polar, 
 and therefore parallel to OG, it will cut OS produced in a point 0' 
 such that OS = 80'.'^ 
 
 Hence the perpendicular from the fixed point 0' on the polar 
 meets it on a fixed line, namely on TSG the common axis of the 
 system of parabolas. 
 
 Therefore the polar envelops a parabola whose focus is 0' and of 
 which the axis of the system of parabolas is the tangent at the 
 vertex. 
 
 5—2 
 
68 THE PARABOLA. 
 
 53. The following are important examples. 
 
 Ex. 1. The locus of the middle points of chords of a parabola 
 which pass through a fixed point is another parabola. 
 
 Let be the fixed point, QQ' any chord through 0. Let ^0 be 
 the diameter through 0. 
 
 Draw the diameter through V the middle point of QQ' cutting the 
 curve in P and the tangent at A in T. Draw AK parallel to the 
 tangent at P cutting PV in K ; and draw PM, KL, VN parallel to 
 the tangent at ^ . 
 
 Then AL = TK=2TP=2AM. 
 
 Also, since AK is parallel to OV, and TFparallel to AN, AL = ON. 
 
 Then VN^=P]iP=4tSA . AM=2SA . AL 
 
 = 2SA. ON. 
 
 Hence the locus of F is a parabola of which ON is a diameter and 
 the tangent at O, the extremity of that diameter, is parallel to the 
 tangent at A. 
 
THE PARABOLA. 69 
 
 Ex. 2. Find the focus and directrix of a parabola which touches 
 four given straight lines. 
 
 Draw the circmn-circles of any two of the triangles formed by 
 taking three of the four given lines, and let S be the point of inter- 
 section of these circles which is not on one of the given lines. Then 
 the feet of the perpendiculars from S on the four given lines will lie 
 in a straight line, and S will be the only point for which this is true. 
 
 Describe a parabola with S as focus, and with the line through 
 the feet of the perpendiculars from S on the given straight lines as 
 the tangent at the vertex; then this parabola will touch the four 
 given straight lines. 
 
 Ex. 3. Find thv focus and directrix of a parabola lohich passes 
 through four given points. 
 
 Let A,B, C,Dhe the four given points, and let AG, BD intersect 
 in 0. 
 
 Take two points X, Y on AC, BD respectively such that 
 
 0Z2 : 07' = AO .OC :B0. OD. 
 
 Then [Prop. XV. Cor.] XY will be parallel to the axis of a 
 parabola through the four given points. Also, if X', Y' be taken 
 on AC, BD respectively so that X'0 = OX and TO=OY, we see that 
 there are two possible directions of the axis of the parabola which 
 are parallel to the lines XY, X' Y respectively. [These lines will be 
 at right angles to one another if the four given points lie on a circle. 
 This follows at once from Art. 44.] 
 
 Let V be the middle point of AD. Draw through V a line 
 parallel to either of the possible axes, and let the line through B 
 parallel to AD cut this diameter in the point W. Then, since AV 
 and B W are ordinates to the diameter VW, 
 
 AV^- :BW^=VP : WP, 
 
 where P is the extremity of the diameter VW. Thus P can be easily 
 found. 
 
 Then the tangent at P is parallel to AD, and the focus S lies 
 on the line through P such that the tangent bisects the angle 
 SPV. 
 
 We can find in a similar manner the extremity of the diameter 
 which bisects another side, and also another straight line on which 
 the focus must lie. 
 
 The focus is thus determined; and, knowing the focus, the 
 direction of the axis, and one point, the directrix is at once found. 
 
70 
 
 THE PARABOLA. 
 
 Ex. 4. Ij the sides of a triangle touch a parabola, the orthocentre 
 of the triangle will be on the directrix of the parabola. 
 
 Let ABC be the triangle whose sides touch a parabola whose 
 focus is S in P, Q, R. Let the directrix cut the sides BC, CA, AB re- 
 spectively in X, Y, Z. Join SZ, and let SZ cut the circle ABCS in 
 C". Join CO' cutting the directrix in and AB in F. 
 
 
 L 
 
 
 / 
 
 ^ 
 
 X- 
 
 /V 
 
 z/ 
 
 ''f 
 
 r 
 
 ^ 
 
 
 aI/^ \ 
 
 "^ 
 
 ^ 
 
 J 
 
 
 
 \ \0 
 
 
 ^ 
 
 /s 
 \ 
 
 
 
 / 
 
 X 
 
 
 \ 
 
 \^ 
 
 
 Then / GO'S- I GAS in the same segment 
 
 = Z SBA 
 
 = complement of Z SZR 
 
 = complement of iFZG'. 
 
 :. GG' is perpendicular to AB. 
 
 But BA bisects the angle /SfZL or OZG'. 
 
 Hence OF=FG'; 
 
 :. L OAB- lFAG'= iBAG'= 1 5 (7F= complement of z GBA 
 
 :. OA is perpendicular to BG. 
 
 Hence is the orthocentre, and it is a point on the directrix. 
 
THE PARABOLA. 
 
 71 
 
 Ex. 5. If a circle touch a parabola at the extremities of a double 
 ordinate PP\ the tangent drawn to the circle from any point on the 
 parabola is equal to the perpendicular distance of that point from the 
 chord of contact. 
 
 The normals at P, P meet on the axis at G ; and the circle with 
 centre G and radius GP will touch the parabola at P and P'. 
 
 If Q be any point on the curve, and QM be the ordinate of <3> 
 then 
 
 = U.S . AM+'iAS^ + 4AS . MN+ MN^ 
 
 = 4:AS {AM+ MN) + 4AS^ + MJtP 
 
 =PN^ + NG^ + MN^ 
 
 = PG^-¥MNK 
 
 But QG2-PG2= square of tangent from Q to the circle whose 
 centre is G and radius GP. 
 
 Hence square of tangent = MtP. 
 
72 THE PARABOLA. 
 
 Ex. 6. The normals at the extremities of any one of a system of 
 parallel chords of a parabola meet on a fixed straight line; also if, 
 the normals at three points on a parabola are concurrent, the circle 
 through the three points will pass through the vertex of the para- 
 bola. 
 
 Let PQ be any one of the system of parallel chords, and let the 
 tangents and the normals at P, Q interisect in the points T, N 
 respectively. 
 
 Let L be the extremity of the diameter through T, and let the 
 tangent at L cut TP, TQ respectively in the points p, q. Then, 
 since the direction of PQ is fixed, i is a fixed point and TL is 
 a fixed diameter. Also p, q are the middle points of TP, TQ 
 respectively. 
 
 P 
 
 Draw pK, qK perpendicular to Tp, Tq respectively. Then, since 
 p, q are the middle points of TP, TQ respectively, it follows that 
 TKNia a straight line and that TK=KN. 
 
 Since TpK and TqK are right angles, TK is a diameter of the 
 circle Tpq. But this circle passes through S, the focus of the 
 parabola. 
 
 Hence TSK is a right angle. 
 
 Also z SKT= L SpT= L SpL + L LPT= L SpL + Z LSp 
 
 - ASLq.. 
 
 But SLq is a constant angle, since X is a fixed point. 
 
 Hence SKT is a constant angle. 
 
 Therefore the right-angled triangle KST is of invariable form, 
 and therafore the ratio SK : ST ia constant. 
 
THE PARABOLA. 73 
 
 Now produce TS to H so that SH= TS. 
 Then z THN= L TBK=x\^\, angle. 
 
 Also TR : HN=2TS : 2SK= const. 
 
 Hence SH : HN is constant. 
 
 But, since 5f is a fixed point such that TS = SH, and the locus of 
 T is a fixed diameter, the locus of H must also be a fixed diameter. 
 
 Then, since SHN is a right angle and SH : HN is constant, the 
 triangle SHN is of invariable form, so that the angle HSN is 
 constant and SN : SH is constant. The locus of N is therefore 
 similar to the locus of H, and we have just seen that If moves on a. 
 fixed straight line. Hence the locus of N is also a straight line. 
 
 Thus the locus of the point of intersection of the normals at the 
 extremities of any one of a system of parallel chords of a parabola is a 
 straight line. [The above proof is given in Taylor's Geometry of 
 Conies, p. 224.] 
 
 Now let A be the vertex of the parabola, and let AR' be parallel 
 to the system of chords, and let R be the other extremity of the 
 ordinate through R, so that AR and AR' are equally inclined to the 
 axis. 
 
 Then, if the normal at R' meet the axis in G, G is one point on 
 the straight line on which all the intersections of normals lie. But 
 G is also on the normal at R ; moreover, if one of the parallel chords 
 be drawn through R, the normals at its extremities must intersect 
 on the normal at R. Hence the normal at R contains two of the 
 points on the straight line on which all the intersections of normals 
 lie; and therefore all the pairs of normals must intersect on the 
 normal at R. Hence the normals at P, Q meet at a point on the 
 normal at R which is such that PQ and AR are equally inclined to 
 the axis of the parabola, whence it follows that P, Q, R, A are 
 cyclic. 
 
 This proves Harvey's Theorem that if the normals at three points 
 on a parabola be concurrent the circle through the three points will 
 pass through the vertex of the parabola. 
 
 ^^ OP Tap. "^ 
 
74 EXAMPLES ON THE PARABOLA. 
 
 EXAMPLES ON THE PAEABOLA. 
 
 1. Shew that, if the length of a chord of a parabola be double the 
 distance of its middle point from the directrix, the chord must pass 
 through the focus. 
 
 2. On a given base AB any isosceles triangle ABP is described, and 
 on AP as base another isosceles triangle APQ similar to the former is 
 described. Shew that the locus of Q is a parabola whose focus is A and 
 whose directrix bisects AB at right angles. 
 
 3. ^ is a fixed point, Q is any point on a fixed straight line, QP is 
 drawn perpendicular to the fixed line, and AP is perpendicular to AQ. 
 Shew that the locus of P is a parabola. 
 
 4. PSP' is a focal chord of a parabola, and the lines through P, P' 
 parallel to the axis meet the normals at P', P respectively in K, K'. 
 Shew that PP'K'K is a rhombus. 
 
 5. A line through the vertex of a parabola perpendicular to the 
 tangent at any point P meets the line through P parallel to the axis in Q. 
 Shew that the locus of Q is a straight line perpendicular to the axis. 
 
 6. The normal at the point P of a parabola cuts the axis in G, and 
 SY is the perpendicular from the focus on the tangent at P. YS is 
 produced to R so that YS = SR. Shew that PYRG is a rectangle, and 
 that the circle PYRG passes through the vertex of the parabola. 
 
 7. Shew that the length of the normal PG at any point P of a 
 parabola is equal to the ordinate which bisects PG. 
 
 8. A is the vertex of a parabola and P is any point on the curve. A 
 line through P perpendicular to PA cuts the axis in G, and Q is taken on 
 GP produced so that GP = PQ. Shew that the locus of Q is a parabola 
 whose focus is A, 
 
 9. Shew that, if two parabolas have a common directrix, the line 
 joining their common points is parallel to and midway between the line 
 joining the points of contact of their common tangents. 
 
 10. A circle is drawn to touch the axis of a parabola, the focal 
 distance SP of any point P, and also the diameter through P ; shew that 
 the centre of the circle must lie on another parabola or on the tangent at 
 the vertex of the original parabola. 
 
 11. AGP is a sector of a given circle whose centre is C, and of which 
 the radius CA is fixed. A circle is drawn touching the arc AP externally 
 and also touching CA, GP produced. Shew that the centre of this circle, 
 for different positions of P, is always on one or other of two parabolas. 
 
 12. Shew that the locus of the point of intersection of the normals 
 to a parabola at the extremities of a focal chord is another parabola. 
 
EXAMPLES ON THE PARABOLA. 75 
 
 13. An endless Btring OPQ is fastened at and two small beads 
 P, Q slide along it ; the string is kept stretched and the beads move so 
 that OP is always equal to OQ and PQ always fixed in direction ; shew 
 that P and Q move on arcs of two parabolas with a common focus and 
 latus-rectum. 
 
 14. P is any point on a parabola whose focus is S and vertex A. 
 The perpendicular from S to AP cuts the tangent at the vertex in R. 
 Shew that the ordinate of P is 4AR. 
 
 15. If perpendiculars be let fall on any tangent to a parabola from 
 two fixed points on the axis equidistant from the focus, the difference of 
 their squares will be constant. 
 
 16. Through any point P of a parabola the chord is drawn which 
 makes an angle with the axis equal to that made by the tangent at P. 
 Shew that the chord touches a fixed parabola. 
 
 17. PiVis the ordinate at any point P of a parabola, and the tangent 
 at P cuts the tangent at the vertex in Y. Shew that NY touches an 
 equal parabola. 
 
 18. From an external point two tangents are drawn to a parabola, 
 and from the points where they cut the directrix two other tangents are 
 drawn meeting the tangents from in A, B. Prove that AB passes 
 through the focus S, and is perpendicular to OS. 
 
 19. From a point P on a circle the ordinate PN is drawn to a fixed 
 diameter AA' and is produced to Q so that the square on PN is equal to 
 the rectangle contained by NQ and a given line ; prove that the locus of 
 Q is a parabola. 
 
 20. TP, TQ are the tangents at the points P, Q of a parabola whose 
 focus is S ; prove that, if SP+SQ be constant, the locus of T is a 
 parabola. 
 
 21. PSP' is any focal chord of a parabola whose vertex is A, and 
 AP, AP' cut the latus-rectum in K, ^'respectively. Shew that, if PN, 
 FN be the ordinates of P, P' respectively, then NPSK' and N'FSK wiU 
 be parallelograms. 
 
 22. TQ, TQ' are tangents to a parabola whose focus is S, and the 
 diameter through T cuts the curve in P ; shew that 
 
 TQ.TQ'=4TP.T8. 
 
 23. If a parabola roll upon an equal parabola which remains fixed, 
 the vertices of the parabolas originally coinciding, the focus of 
 the moving parabola describes the directrix of the fixed parabola, and 
 the latus-rectum and the tangent at the vertex of the moving para- 
 bola touch fixed circles. 
 
 24. Two parabolas have a common focus and from any point on 
 their common tangent are drawn the other tangents to the two parabolas. 
 Prove that the angle between these tangents is equal to the angle between 
 the axes of the conies. 
 
 25. Shew that, if TP, TQ are tangents to a parabola, PQ being the 
 normal at P; then will TQ be bisected by the line through S per- 
 pendicular to TS. 
 
76 EXAMPLES ON THE PARABOLA. 
 
 26. Any number of equal parabolas have parallel axes and a 
 common point ; shew that their vertices lie on a parabola whose vertex 
 is at the given point. 
 
 27. The latus-rectum of a parabola is given, also the direction of 
 its axis and a fixed point on the curve. Shew that the locus of its 
 focus is a parabola. 
 
 28. Shew that all parabolas which have a given directrix and a 
 given tangent at a given point, will touch a fixed parabola of which the 
 given point is focus. 
 
 29. Shew that all parabolas which have a common directrix, and 
 whose foci are on a fixed circle, touch two fixed parabolas. 
 
 30. Prove that the locus of points at which two parabolas with the 
 same focus subtend equal angles is the straight line bisecting the angle 
 between their directrices. 
 
 31. If the triangle formed by three tangents to a parabola be 
 isosceles, the line joining the vertex of the triangle to the focus will pass 
 through the point of contact of the base. 
 
 32. If an equilateral triangle circumscribe a parabola, the lines 
 from the focus to the angular points of the triangle will pass through 
 the points of contact. 
 
 33. If a quadrilateral be inscribed in a circle one of its three 
 diagonals will pass through the focus of the parabola which touches its 
 sides. 
 
 34. Normals at P, p the extremities of a focal chord of a parabola 
 intersect the axis in G, g. Shew that the line drawn through the 
 middle point of Pp perpendicular to Pp will pass through the middle 
 point of Gg. 
 
 35. Shew that two parabolas with parallel axes can only intersect in 
 two points. 
 
 36. The normals at P, p the extremities of a focal chord of a 
 parabola intersect the axis in G, g respectively, and the tangents at P, p 
 meet in T. Shew that the circles SPG, Spg intersect at a point R on TS 
 produced such that TS — SB. 
 
 37. If -4J5C be a triangle inscribed in a parabola, and A'B'C be a 
 triangle formed by three tangents parallel to the sides of the triangle 
 ABC, then will the sides of ABC be four times the corresponding sides of 
 A'B'C'. 
 
 38. If the tangent at a point P of a parabola meet the axis in T, 
 and the chord PQ and the tangent PT make equal angles with the axis ; 
 thenPQ=4Pr. 
 
 39. If two tangents be drawn to a parabola, the perpendicular from 
 the focus on their chord of contact will pass through the middle point of 
 their intercept on the tangent at the vertex. 
 
EXAMPLES ON THE PARABOLA. 77 
 
 40. If the ordinate PN at any point P of a parabola be produced to 
 Q such that PQ = SP; prove that the locus of Q is a parabola which 
 touches the tangent at the vertex of the original parabola, the corre- 
 sponding diameter being the tangent at the extremity of the latus-rectum 
 of the original parabola. 
 
 41. The tangent at a variable point P on a parabola riieets the 
 tangent at a fixed point Q in T and FT is divided in a fixed ratio in R. 
 Prove that the locus of P is a parabola touching the given parabola at Q. 
 
 42. Shew that the envelope of the directrices of all parabolas which 
 have a common vertex A and pass through a fixed point P is a parabola 
 the length of whose latus-rectum is AP. 
 
 43. Inscribe in a given parabola a triangle having its sides parallel 
 to three given straight lines. 
 
 44. If through a fixed point A a straight line be drawn meeting two 
 fixed lines OD, OE in B and G respectively, and on it a point P be taken 
 such that AG .AP=AB^', prove that the locus of P is a parabola which 
 passes through A and 0, and has its axis parallel to OD^ and the tangent 
 at A parallel to OE. 
 
 45. Inscribe a circle in a segment of a parabola cut off by a double 
 ordinate. 
 
 46. A. point V is taken on an ordinate PM, produced, of a parabola, 
 and ME is taken on MP a mean proportional between PM and MV ; the 
 diameters through E, V meet the curve in R, Q respectively. Prove that 
 PQ meets the axis in the foot of the ordinate of R. 
 
 47. Focal chords of a parabola at right angles to one another meet 
 the directrix in T, T'. Shew that the bisectors of the angles between the 
 tangents from either T or T' are parallel to the tangents from the other. 
 
 48. ^PC is any isosceles triangle described on a given base AB, and 
 the tangents at ^, C to the circle ABC intersect in P. Shew that the 
 locus of P is a parabola whose focus is A, whose axis is along AB and 
 whose latus-rectum is equal to AB. 
 
 49. If a leaf of a book be folded over so that one of the outer corners 
 is on the inner side, the line of the crease will envelop a parabola whose 
 directrix is the inner side. 
 
 50. Shew that a line which cuts two fixed lines OA, OB in the 
 points P, Q respectively so that OP+OQ is constant, touches a fixed 
 parabola. 
 
 51. Shew that a line which cuts two fixed straight lines so that the 
 difference of the intercepts, measured from their point of intersection, is 
 constant envelops a parabola. 
 
 52. In a given regular polygon is inscribed any regular polygon of 
 the same number of sides ; prove that the envelope of each side is a 
 parabola. 
 
 53. A line cuts two given circles so that the intercepted chords are 
 equal ; shew that the line envelops a parabola. 
 
78 EXAMPLES ON THE PARABOLA. 
 
 54. Shew that all chords of a parabola, which have their middle 
 points on a line perpendicular to the axis of the parabola, will touch 
 another parabola. 
 
 55. If a chord of a parabola subtend a right angle at the vertex it 
 will cut the axis at a distance from the vertex equal to the latus-rectum. 
 
 56. QQ' is any chord of a parabola ; any diameter cuts the tangent 
 at Q in T, the curve in F and the chord QQ' in K; shew that 
 
 TP:PK=QK:KQ'. 
 
 57. Draw through a given point within a parabola a chord which 
 shall be divided in a given ratio at the point. 
 
 58. The tangent at any point P of a parabola meets the tangents 
 drawn from any point O in A, B and the diameter through O in G. 
 Prove that ^P=(7£. 
 
 59. Any chord PQ of a parabola cuts the axis in O ; shew that 
 
 PQ'^=AP^ + AQ^ + 2RO'^-2AO\ 
 A being the vertex, and RO the ordinate through O. 
 
 60. T is the intersection of the tangents at the ends of any chord of 
 a parabola, T' a corresponding point for any perpendicular chord ; shew 
 that the rectangle of the abscissae of T and T is equal to the rectangle 
 of the intercepts of the chords on the tangent at the vertex. 
 
 61. Having given a point, a tangent and the direction of the axis of 
 a parabola, shew that the locus of its focus is another parabola. 
 
 62. Two parabolas are described touching the sides of a triangle and 
 having their foci at the extremities of a diameter of the circum-circle ; 
 shew that their axes intersect on the circum-circle and that the tangents 
 at their vertices intersect on the nine-point circle of the triangle. 
 
 63. Through two fixed points ^, ^ on the axis of a parabola two 
 chords PAQ, PBB are drawn, and QR cuts the axis in T; shew the 
 ratio TQ : TR is independent of the position of P. 
 
 64. The locus of the centre of a circle which cuts a given straight 
 line and a given circle so that the chords are equal and of constant length 
 is a parabola. 
 
 65. The tangent at a point P of a parabola makes the same angle 
 with the axis as the line drawn from the focus to the point of intersection 
 of the tangents at two points Q and R. Shew that this relation is sym- 
 metrical and that the circum-circle of the triangle formed by the three 
 tangents will touch the axis of the parabola. 
 
 66. P'S'P' is a focal chord of a parabola ; V is the middle point of 
 PP', and VG is drawn perpendicular to PSP' to cut the axis in G. 
 Shew that SG, VG are the arithmetic and geometric means between SP 
 and SP'. 
 
 67. 04, OB are tangents to a parabola; shew that the circles 
 passing through O and touching AB oi A and B respectively will 
 intersect on the diameter through O, and will have their centres on the 
 directrix. 
 
EXAMPLES ON THE PARABOLA. 79 
 
 68. Any circle whose centre is at a given point cuts two fixed parallel 
 straight lines in the points A, A' and B, B' respectively. Shew that the 
 lines AB, AB', A'B, A'B' will all touch a fixed parabola. 
 
 69. -P is any point on a parabola whose focus is S, and PS is 
 produced to P' so that PS = SP'. Also P'Q, P'B are the tangents from P' to 
 the parabola. Shew that the circle PQR touches the parabola at P and 
 passes through P'. 
 
 70. Two parabolas have a common focus and their axes in opposite 
 directions ; prove that the locus of the middle points of chords of either 
 which touch the other is another parabola. 
 
 71. The three middle points of the diagonals of any quadrilateral 
 are on a straight line parallel to the axis of the parabola which touches 
 the sides of the quadrilateral. 
 
 72. Two equal and similarly situated parabolas have the same axis, 
 and a tangent is drawn at any point Q of the inner which cuts the outer 
 in the points P, P'. Shew that Q, is the middle point of PP', and that 
 the distance between the diameters through P and P' is constant for all 
 positions of Q. 
 
 73. Two confocal and coaxial parabolas are cut by a line parallel to 
 their axis in the points P, P', and the tangents at P, P' intersect in T. 
 Shew that T is midway between their directrices, and that TS bisects the 
 exterior angle PSP'. 
 
 74. The tangents at two points one on each of two confocal and 
 coaxial parabolas meet in T\ shew that if T be equidistant from the 
 diameters through the points of contact of fhe tangents it will be also 
 equidistant from the directrices of the parabolas. 
 
 75. -P-P'j QQ' are the chords of contact of pairs of tangents from a 
 point to two confocal and coaxial parabolas ; shew that if P, Q, S are 
 in a straight line, so also will P', Q', S be, and that PQ' and QP' will 
 be parallel to the axis. 
 
 76. If a circle through the focus of a parabola touch the curve at P 
 and cut it at L and M, and the circle cut the axis in N ; then will 
 LP=MN. 
 
 77. PQ is a chord of a parabola normal at P ; QR is drawn parallel 
 to the axis to meet the double ordinate PP' produced in B. Prove that 
 the rectangle contained by PP' and P'B is constant. 
 
 78. TA, TA' are two tangents to a parabola and P is any other 
 point on the curve; the tangent at P meets the diameters through A, A' 
 in a, a' respectively; also lines through P parallel to the tangents at A, 
 A' meet the diameters through A', A respectively in Q', Q. Shew that 
 QQ' is parallel to the tangent at P, and that, if O be the pole of QQ'^ Oa, 
 Oa' are parallel to the tangents at A', A respectively. 
 
 79. A triangle ABC is formed by three tangents to a parabola, 
 another triangle DEF is formed by joining the points in which the 
 chords through two points of contact meet the diameter through the 
 third. Shew that A, P, G are the middle points of the sides of DEF. 
 
80 EXAMPLES ON THE PARABOLA.. *i 
 
 80. A circle is described on a chord AB of a parabola as diameter, 
 and cuts the curve again in (7, D. Prove that, if the chord AB be fixed 
 in direction, the difference of the squares of AB and CD is constant. 
 
 81. The tangent at any point P of a parabola is cut by any two 
 other tangents in the points B, R', and the diameter through P is cut by 
 the chord of contact of those tangents in ; shew that RP . PR' = SP . PO, 
 S being the focus. 
 
 82. TQf TR, tangents to a parabola, meet the tangent at P in Z 
 and Y respectively, and TU is drawn parallel to the axis, meeting the 
 parabola in U. Prove that the tangent at U passes through the middle 
 point of XY, and that, if S be the focus, 
 
 XY-^ = 4:SP . TU. 
 
 83. Two fixed tangents are met by a variable tangent in the points 
 X, Y. Shew that, if a chord of the parabola be drawn equal and parallel 
 to XY, it will envelop an equal parabola. 
 
 84. If PQ be a focal chord of a parabola and R any point on the 
 
 P2J2 
 
 diameter through Q ; shew that the focal chord parallel to PJR=-p^. 
 
 85. Through any point on a parabola two chords are drawn equally 
 inclined to the tangent there. Shew that their lengths are proportional 
 to the portions of their diameters intercepted between them and the 
 curve. 
 
 86. PP' is a focal chord of a parabola, and the normals at P, P' 
 meet the curve again in Q, Q'. Shew that QQ' is parallel to PP'. 
 
 87. Shew that the locus of the focus of a parabola which touches 
 two given straight lines and whose directrix passes through a given 
 point, is a circle. 
 
 88. OQ, OQ' are two tangents to a parabola, and the diameter 
 through cuts the curve in P and QQ' in V ; shew that, if the tangent 
 at P cut OQ, OQ' respectively in R, R', then will QR', Q'R be divided by 
 the parabola in the ratio 8 to 1. 
 
 89. Two circles are drawn each touching a parabola at the extremi- 
 ties of a double ordinate. Shew that the sum or the difference of the 
 lengths of the tangents drawn to these circles from any point on the 
 parabola is constant and equal to the distance between the chords of 
 contact. 
 
 90. Two circles have double contact with a parabola; shew that 
 their radical axis is midway between their chords of contact. 
 
 91. -P, Q, -R, S are any four points on a parabola; PQ meets the 
 diameter through R in R' and RS meets the diameter through Q in Q' ; 
 shew that Q'R' is parallel to PS. 
 
EXAMPLES ON THE PARABOLA. 81 
 
 92. The polar of the middle point of a normal chord of a parabola 
 meets the focal vector to the point of intersection of the chord and the 
 directrix on the normal at the further end of the chord. 
 
 93. The axes of the two parabolas which pass through four given 
 points on a circle will intersect at right angles in the centroid of the four 
 points. 
 
 94. Shew that, if two parabolas intersect in four points on a circle, 
 their axes must be at right angles ; and conversely, that if the axes of 
 two parabolas be at right angles and they intersect in four points, the 
 four points will be cyclic. 
 
 95. A variable tangent to a given parabola cuts a fixed tangent in 
 the point P ; shew that the line through P perpendicular to the variable 
 tangent envelops another parabola. 
 
 96. On any chord of a parabola as diameter a circle is described 
 cutting the parabola in two other points. Shew that the portion of the 
 axis intercepted between the original chord and the chord joining the 
 other two points is equal to the latus-rectum. 
 
 97. If a triangle be inscribed in a parabola, the points where the 
 sides produced meet the tangents at the opposite angles will lie on a 
 straight line. 
 
 98. The ratio of the segments into which any tangent to a parabola 
 is divided by three fixed tangents is constant. 
 
 99. If TP, TQ be two tangents to a parabola, the perpendicular 
 distance of T from any other tangent will be a mean proportional 
 between the perpendicular distances of P and Q from the same tangent. 
 
 100. If four tangents be drawn to a parabola, and if A and A', B 
 and B', G and C be the extremities of the three diagonals of the quadri- 
 lateral formed by the tangents; then will the product of the perpen- 
 diculars from A and A' on any tangent, be equal to the product of the 
 perpendiculars from B and B' or from G and C" on that tangent. 
 
 S. O. 
 
CHAPTER III. 
 The Ellipse. 
 
 54. From the definition of an ellipse as the locus of a 
 point which moves in the plane containing a given point, 
 called the focus, and a given straight line, called the 
 directrix, in such a manner that its distance from the 
 focus is always in a constant ratio to its perpendicular 
 distance from the directrix, we have already proved in 
 Chapter I. that an ellipse is symmetrical about the line 
 through its focus perpendicular to its directrix ; and also 
 that, if this line cut the curve in the points A, A\ and G 
 be the middle point of AA', the ellipse is symmetrical 
 about the line through C parallel to the directrix, whence 
 it follows that there is a second focus in the line A A' and 
 a second directrix perpendicular to AA'. 
 
 We have also proved that, if S, S' be the two foci, and 
 if A A' produced cut the directrices in the points X, X' 
 respectively, then 
 
 CB;GA = a A :CX = SA'. AX. 
 
THE ELLIPSE. 
 
 83 
 
 The terminated lines A A and BB' are called respec- 
 tively the major and the minor axes of the ellipse. 
 
 
 
 
 B 
 
 
 \ 
 
 M 
 
 / 
 
 I 
 
 "7 
 
 
 X 
 
 A 
 
 V 
 
 I 
 
 s 
 
 G 
 
 S' 
 
 / 
 
 Let BM be the perpendicular from B on the directrix. 
 Then 
 
 m :BM=SA :AX=GA : CX. 
 
 Hence, as BM = GX, SB = GA. 
 
 Therefore BG' = BS' - GS' = GA' - GS\ 
 
 Now GA' - GS' = (GA - GS) (GA + GS) 
 
 = AS.SA\ since GA = GA\ 
 
 Also GA' - GS' = GS,GX-GS' = GS. SX. 
 
 Thus BC^ = CA2 - CS2 = AS . SA' = CS . SX. 
 
 Again, if LSL^ be the latus-rectum, 
 
 SL : SX = SA : AX = GS : GA ; 
 
 .-. SL,CA=CS.SX = BC2. 
 
 Thus the semi-minor axis is a mean proportional to the 
 semi-major axis and the semi-latus-rectum. 
 
 6—2 
 
84 
 
 THE ELLIPSE. 
 
 55. Prop. I. The sum of the focal distances of any 
 point on an ellipse is constant 
 
 Let S, S' be the foci and P be any point on the ellipse. 
 Join SP, ST, and draw through P the line MPM' per- 
 pendicular to and terminated by the directrices. 
 
 Then SP:MP=GA:CX; 
 
 and ST:PM' = GA:GX; 
 
 .'. SP-\-S'P'.MP + PM = CA'.GX. 
 
 But MP-\-PM = MM=^XX'=WX. 
 
 Hence /SfP+^T = 20A 
 
 It can easily be proved that, if Q be any point external to the 
 ellipse, SQ + S'Q is greater than 20 A. 
 For, ii SQ cut the ellipse in P, 
 
 SQ + S'Q = SP+PQ + S'Q>SP+PS\ 
 Similarly, if Q be an internal point, SQ + S'Q<:2CA. 
 
 The above property of an ellipse enables us to describe 
 the curve by the continuous motion of a point. 
 
 For, if the ends of a string of finite length be fastened 
 at two points S, S\ and the string be kept tight by a 
 pencil moving along it ; then, if P be any position of the 
 point of the pencil, SP + S'P will be constant and equal 
 to the length of the string, and therefore P will be on an 
 ellipse whose foci are S, S' and whose major axis is equal 
 to the length of the string. 
 
THE ELLIPSE. 
 
 85 
 
 56. Prop. II. The tangent at any point of an ellipse 
 is equally inclined to the focal distances of the point. 
 
 Let >§, S' be the foci of the ellipse, and let P be any 
 point on the curve. 
 
 Draw through P the line MPM' perpendicular to and 
 terminated by the directrices. 
 
 Let the tangent at P cut the directrices in K, K', 
 
 K' 
 
 Join /SfP, ST, SK and S'K\ 
 Then, from the similar triangles MPK, MTK\ 
 KP:PK' = MP'.PM' 
 =:SP:ST. 
 Also we know [Art. 12] that the angles KSP and 
 K'S'P are right angles. 
 
 Hence the triangles KPS^ K'PS' are similar, and 
 therefore 
 
 ZSPK= ZSTK\ 
 Thus the tangent at P to the ellipse bisects the angle 
 between SP and S'P produced. 
 
 Since the normal is perpendicular to the tangent, it 
 follows that the norwul bisects the angle SPS\ 
 
86 
 
 THE ELLIPSE. 
 
 57. Prop. III. If the tangent at any point P of an 
 ellipse cut the major axis GA produced in T, and PN he 
 the perpendicular on the aosis ; then will CN . CT — GA\ 
 
 Draw MPM' parallel to the axis-major cutting the 
 directrices in the points M, M'. 
 
 Then, since TP bisects the exterior angle SPS\ 
 ST:S'T = SP:ST 
 
 = PM:MT = NX iX'JST; 
 
 i.e. 2CT:2CS=2GX:2GK 
 
 Hence GT.GN=GS.GX = GA\ 
 
 58. Prop. IV. If the tangent at any point P of an 
 ellipse meet the minor axis GB produced in t, and Pn he 
 the perpendicular on the axis ; then will Gn.Gt = GB^. 
 
 Let Pn produced cut the curve again in P\ and the 
 directrix in M. Then, since every chord perpendicular to 
 the minor axis is bisected by that axis, it follows, as in 
 Art. 18, Cor. II., that the tangents at P, P' will meet on 
 the minor axis, and therefore at the point t. 
 
THE ELLIPSE. 
 
 87 
 
 Then, we know [Art. 10 and Art. 17] that SM and St 
 are respectively the external and the internal bisectors 
 of the angle PSP\ and are therefore at right angles. 
 
 Hence Z CSt = complement of / X8M 
 
 'Oti^rt 
 
 The right-angled triangles GSt and XMS are therefore 
 similar, and we have 
 
 Ct:GS==SX: XM =SX:Gn; 
 .'. Gt.Gn = GS.8X^BG\ 
 
 EXAMPLES. 
 
 1. Shew that, if a focus of an ellipse, the length of its major axis, 
 and one point on the curve be given, the locus of the centre will be a 
 circle. 
 
 2. One focus of an ellipse and the corresponding directrix are given, 
 and it is also known that a given straight line touches the ellipse. Find 
 the other focus. 
 
 3. Find the locus of the centre of an ellipse which has a given focus 
 and which touches a given straight line at a given point. 
 
 4. A number of ellipses have a common major axis, and are cut by 
 any line perpendicular to that axis. Shew that the tangents at all the 
 points of section will meet in a point on the major axis. 
 
 5. The tangent and the ordinate of any point P on an ellipse cut the 
 major axis CA in the points T and N respectively. Shew that NA is less 
 than AT. 
 
88 
 
 THE ELLIPSE. 
 
 59. Prop. V. If the normal at a point P of an 
 ellipse cut the major and minor axes respectively in the 
 points Gy g ; then will the ratio PG : Pg be constant. 
 Also, if PJSF and Pn he the perpendiculars on the aods 
 major and axis minor respectively, then will the ratios 
 GG : GN and Gg : Gn he constant 
 
 Join P to the foci B, 8' of the ellipse. Draw through 
 P the line MPM' perpendicular to and terminated by 
 the directrices. Then, since MM' and SS' are both 
 bisected by the minor axis, it follows that M8, M'8' 
 when produced will meet on the minor axis. Let them 
 meet at the point g, and let Pg cut the major axis in the 
 point G. 
 
 Then SG :MP = Gg: Pg = G8' : PM'; 
 
 .: SG: GS' = MP:PM' 
 
 = SP:PS\ 
 
 Hence PGg bisects the angle SPS\ and must there- 
 fore [Prop. II.] be the normal at P. 
 
THE ELLIPSE. 89 
 
 Then from similar triangles, 
 
 PG:Fg = MS : Mg = XS : XG ', 
 .'. PG:Pg = GS.SX:GS.GX=GB':GA\ 
 Again GG : GN= GG : nP=Gg : Pg 
 
 = Sg:Mg 
 = GS : GX. 
 Also Gg:Gn = gG:GP=gS:SM 
 
 = GS : SX. 
 
 60. Prop. VI. The circle through the foci of an 
 ellipse and any point P on the curve passes through the 
 points in which the minor axis is cut hy the tangent and 
 normal at P. 
 
 Let the circle SPS' cut the minor axis in the points 
 ty g ; then these points will clearly be on opposite sides of 
 SS'. Let P and g be on opposite sides of SS'. 
 
 Then, since tg bisects SS' at right angles, it is the 
 diameter of the circle, and the arcs Sg, S'g must be equal, 
 whence the angles SPg, S'Pg are equal. Hence Pg is 
 the normal at P, and since tgis the diameter, of the circle, 
 tPg is a right angle, and therefore Pt is the tangent at P 
 to the ellipse. 
 
 Cor. Since >Si, S\ g, t are on a circle, 
 gG,Gtr=:S'G.GS=GS\ 
 Also, since the triangles GGg and tGT are similar, 
 GG:gG=Gt:GT. 
 Thus GG. GT = gG. Gt = S'G. GS= OS\^,^sffffSSSsssss^^ 
 
 ^^^ or THS ^^ 
 
 pririVBRSiTi 
 
90 THE ELLIPSE. 
 
 61. Prop. VII. Shew that the feet of the perpendicu- 
 lars from the foci on the tangent at any point of an ellipse 
 lie on a fixed circle, and that the semi-minor axis is a 
 mean proportional to the lengths of the perpendiculars. 
 
 Let BY, S'Y' be the perpendiculars from the foci on 
 the tangent at any point P of an ellipse. 
 
 Join >SfP, ST and let ST, SY produced meet in H. 
 Join GY. 
 
 Then ZSPY= zSTY'= Z.HPY. 
 
 Also Z. SYP = rt. angle = Z HYP, 
 
 and PF is common to the two triangles SPY, HPY 
 
 Hence the two triangles SPY, HPY are equal in all 
 respects, so that 
 
 SY = YH 8^nd SP = PH. 
 
 Hence S'H = ST •\-PH= ST + PS= 2GA. 
 
 But, since SH = 2SY and SS' = 2SC, 
 
 it follows that GY is parallel to ST, and 
 
 2GY = S'H=2GA. 
 
THE ELLIPSE. 91 
 
 Thus GY= CA, and therefore the point Y is on the 
 circle whose centre is G and radius GA. 
 
 It can be proved in a similar manner that GY' is 
 parallel to SP and equal to GA. 
 
 Thus the feet of the perpendiculars from the foci on 
 any tangent to an ellipse are on the circle of which the 
 major axis is a diameter. 
 
 Def. The circle of which the major axis of an ellipse 
 is a diameter is called the auxiliary circle. 
 
 Now produce Y'G to meet the auxiliary circle again 
 in the point Z. Then, since Y'GZ is a diameter of the 
 auxiliary circle, the angle Y^YZ is a right angle. The 
 angle Y^YS is also a right angle, and therefore F>Si^is a 
 straight line. 
 
 Since GZ = GY\ GS = GS' and Z SGZ = Z SVY', 
 
 it follows that the triangles SGZ, B'GY' are equal in all 
 respects, so that S'Y' = SZ. 
 
 Hence /SfF./Sf'F = /SfF. ZS= A8.SA' = BG\ 
 
 Cor. I. If a line through G parallel to the tangent at 
 P cut PS, PS', produced if necessary, in the points E, E' 
 respectively ; then PE = PE' = AG. 
 
 For, CE is parallel to P¥' and GT parallel to PSE ; hence 
 PE = GT= GA. SimUarly PE'= GY= GA. 
 
 Cor. II. The rectangle contained by the perpendiculars 
 from a focus on two parallel tangents to an ellipse is 
 constant 
 
 The converse of Prop. vii. is important, namely that 
 if S be any point within a given circle and S be joined to 
 any point Y on the circle, then the line through Y perpen^ 
 dicular to YS will always touch the ellipse of which S 
 is a focus and of which the given circle is the auxiliary 
 circle. 
 
92 THE ELLIPSE. 
 
 EXAMPLES. 
 
 1. The portion of the mmor axis intercepted between the tangent 
 and the normal at any point of an ellipse can never be less than the 
 distance between the foci. 
 
 2. Shew that PG touches the circle SPM. 
 
 3. Shew that the circles PSM, PS'M' touch one another. 
 
 4. Shew that the triangles PSg, MPg are similar, and that the ratio 
 Sg : Pg is constant. 
 
 5. Shew that the triangles GSP, gS'P are similar, and that 
 
 PG.Pg = SP.S'P. 
 
 6. Shew that the triangles SPT^ tPS' are similar, and that 
 
 SP.S'P=TP.Pt. 
 
 7. Draw a tangent to an eUipse parallel to a given straight line. 
 
 8. Find the foci of an ellipse, having given the tangent at a given 
 point and also the auxiliary circle. 
 
 9. Construct an ellipse having given the foci and one tangent. 
 
 10. Construct an ellipse having given three tangents and one focus. 
 
 11. Shew that the circle on SP as diameter will touch the auxiliary 
 circle. 
 
 12. Any tangent to an ellipse is cut by the tangents at its vertices in 
 the points T, T'. Shew that the circle on TT' as diameter will pass 
 through the foci. 
 
 13. Shew that YEE'Y' is a parallelogram. 
 
 14. Shew that SY' and S'Y intersect in the middle point of PG. 
 
 15. Shew that the circle YGY' passes through the foot of the ordi- 
 nate of P. 
 
 16. Shew that PN bisects the angle YNY\ 
 
 17. SY, SZ are the perpendiculars from a focus on the tangent and 
 r.ormal at any point of an eUipse. Shew that YZ passes through the 
 centre of the ellipse. 
 
 18. An ellipse of given eccentricity has a given focus and touches a 
 given straight line. Shew that its centre is on a fixed circle. 
 
 19. Two ellipses have a common focus and equal minor axes ; shew 
 that their common tangents are parallel. 
 
 20. Two pairs of parallel tangents are drawn to an ellipse and 
 parallels to them are drawn through a focus ; shew that the four points 
 where the parallels through the focus meet the tangents lie on a circle. 
 
THE ELLIPSE. 93 
 
 62. Prop. VIII. The point of intersection of two 
 tangents to an ellipse which are at right angles to one 
 another is on a fixed circle. 
 
 Let T be the point of intersection of two perpendicular 
 tangents. Draw BY, S'Y' perpendicular to one tangent 
 and SZ, S'Z' perpendicular to the other. 
 
 Then TZ. TZ' = SY.S'r = BG\ 
 
 Hence, as Z, Z' are on the auxiliary circle, the square of 
 the tangent from T to the auxiliary circle is equal to BG'^. 
 
 Hence GT'-GA' = GB'. 
 
 Thus T is on the circumference of the circle the square of 
 whose radius is equal to ^0^+ BC^. 
 
 Def. The circle which is the locus of the point of 
 intersection of perpendicular tangents to an ellipse is 
 called the Director Circle. 
 
 Ex. 1. The director-circle of an ellipse cannot cut the directrix in 
 real points. 
 
 Ex. 2. The length of the tangent to the director-circle of an ellipse 
 drawn from any point on the directrix is equal to the distance of that 
 point from the focus. 
 
 Ex. 3. Find the locus of the centre of an ellipse whose axes are of 
 given length , and which touches two fixed perpendicular straight lines. 
 
94 
 
 THE ELLIPSE. 
 
 63. Prop. IX. If FN he the perpendicular from 
 any point P of an ellipse on the major axis AA\ then will 
 the ratio PN"" : AN. NA' he constant. 
 
 Let PA, PA' produced cut a directrix in the points 
 K, K' respectively. Join K, K' to the corresponding 
 focus S. 
 
 Then KS bisects the angle PBA', and K'S bisects the 
 angle PSA [Art. 10]. Hence KS and K'S are at right 
 angles, and therefore 
 
 K'X.XK^XS\ 
 Now, from similar triangles, 
 
 PN:NA'^K'X:XA' 
 and PN'.AN=XK:XA, 
 
 .-. PN''.AN,NA'==K'X.XK'.XA'.XA 
 = XS':XA.XA'. 
 Thus the ratio PN'^ : A N.N A' is constant for all posi- 
 tions of P. If P be taken at an extremity of the minor 
 axis, PN will be BG and ANNA' will be AC; thus the 
 constant ratio must be equal to BO" : AC^. 
 Hence PN : AN.NA' = BG' : AG\ 
 
THE ELLIPSE. 
 
 95 
 
 Cor. If Pn be the perpendicular from P on the 
 minor axis, then NP = On and Pn — NO ; 
 also AN. NA' = GA^ - ON' = GA' - Pn\ 
 
 Hence Gn' : GA' - Pn" = BG^ : GA\ \ 
 
 or Gn" : BG' = GA^ - Pn' : GA' ; 
 
 .-. BG' - Gn' : BG' = Pn^ : GA'. 
 Hence Pri^ : BG' - Gn' = (7^^ . ^(72 
 
 64. Prop. X. If PN be the ordinate of any point P 
 on an ellipse and NP produced cut the auxiliary circle in 
 the point p ; then pN : PN = BG : GA. 
 
 For, since 
 
 and 
 
 PN 
 
 pN^ 
 
 AN,NA' = BG''.GA\ 
 Gp^-GN^=GA'-GN' 
 = AN.NA\ 
 it follows that PN' : pN = BG' : GA', 
 
 or PN:pN=^BG:GA. 
 
 If the ordinate of an ellipse at a point P is produced 
 to meet the auxiliary circle in the point p, the points P, p 
 are called corresponding points. 
 
96 THE ELLIPSE. 
 
 The tangents at corresponding points on an ellipse and 
 its auxiliary circle intersect on the major axis. 
 
 For let P, P' be any two points on an ellipse, and let 
 p, p' be the two corresponding points on the auxiliary 
 circle. Let PP' cut the major axis in K. 
 
 Then NK : N'K = NP : N'P' 
 
 = Np'.N'p', 
 whence it follows that Kpp is a straight line. 
 
 Now let N'P'p' move up to and ultimately coincide 
 with NPp and it follows that the tangents at P, p will 
 intersect on the major axis. 
 
 The circle on the minor axis as diameter is called the minor 
 auxiliary circle. The properties of this circle are not however 
 of much importance. 
 
 If the perpendicular Pn on the minor axis cut the minor auxi- 
 liary circle in the point Q, it follows from Prop. IX. Cor. that 
 Pn : Qn=AG : BC. 
 
 65. If a line be drawn through P parallel to Cp and cutting the 
 major and minor axes respectively in the points H, K. Then 
 pCKP will be a parallelogram and therefore PK=Cp=CA. Also 
 PH:PN=pC:pN,orPH:pG = PN:pN=BC:CA. B.encePH=BG. 
 Thus PH, PK and HK are all of constant length. 
 
 Conversely, if a line HK of constant length have its extremities 
 on two fixed perpendicular lines, any other fixed point P on the 
 line, or the line produced, will describe an ellipse whose semi-axes 
 are equal to KP, HP respectively. This is the principle of the 
 Elliptic Compasses. 
 
 EXAMPLES. 
 
 1. QQ' is one of a system of parallel chords of a circle, and P is a 
 point on QQ' such that QP : PQ' is constant. Shew that the locus of P 
 is an ellipse. 
 
 2. Find the foci of an ellipse, having given the extremities of the 
 major axis and one point on the curve. 
 
 3. Shew that, if NP be the ordinate of any point on an ellipse, 
 the locus of the middle point of NP is another ellipse. 
 
 4. Shew that, if PP' be any chord of an ellipse parallel to either 
 axis, and Q be the point on PP' such that PQ : QP' is a given ratio, the 
 locus of Q will be another ellipse. 
 
 5. Q is any point on a given circle, and Q3I is the perpendicular 
 drawn from Q on a fixed tangent to the circle. Shew that, if P be the 
 middle point of QM, the locus of P will be an ellipse. 
 
THE ELLIPSE. 
 
 97 
 
 66. Prop. XI. If TQ, TQ' he tangents to an ellipse 
 whose foci are S, S\ the angles STQ, 8'TQ' will he equal. 
 
 Draw from the foci BY, &Y' perpendicular to TQ, 
 and SZ, B'Z' perpendicular to TQ ] and join F, Z and 
 F, Z\ 
 
 Then Z YSZ = Z YS'Z\ since each is supplementary 
 to QTq, 
 
 And, since BY. B'Y = BG^ = BZ . B'Z\ 
 BY:BZ=:B'Z':B'Y. 
 
 Hence the triangles YBZ, Z'B'Y are similar, and 
 
 ^BZY= /.B'Y'Z'. 
 
 But B, F, T, Z are cyclic, since BYT and BZT are 
 right angles ; 
 
 .'. the angles BTY and >Sf^Fare equal or supplementary. 
 
 Similarly the angles B'TZ' and B'Y'Z' are equal or 
 supplementary. 
 
 Hence Z BTQ = Z >SfTQ', for they obviously cannot be 
 supplementary. 
 
 Since Z QTB = Z QT>Sf; it follows that the internal 
 and external bisectors of the angle BTB' are also the 
 internal and external bisectors of the angle QTQ\ 
 
 s. c. 7 
 
98 THE ELLIPSE. 
 
 Properties of Diameters. 
 
 67. We have already proved that the locus of the 
 middle points of any system of parallel chords of a conic 
 is a straight line through the centre of the conic, which is 
 called a diameter of the conic. We have also proved that 
 the tangents at the extremities of any chord intersect on 
 the diameter which bisects the chord. 
 
 Now it is obvious that any line through the centre of 
 an ellipse must cut the curve in two real points ; and we 
 have proved in Art. 18, that the tangents at the extremi- 
 ties of any diameter are parallel to the system of chords 
 bisected by that diameter. 
 
 68. Prop. XII. If the diameter PGP' bisect all 
 chords of an ellipse parallel to the diameter BCD', then 
 mill the diameter DGB' bisect all chords of the conic which 
 are parallel to PGP'. 
 
 Let QQ be any chord of the ellipse parallel to DGD' ; 
 then F, the middle point of QQ' is on PGP*. 
 
 Join Q'G and produce it to cut the ellipse again in c[. 
 Join Qq' cutting BGU in W. 
 
 Then, since Q'Q is bisected in V and Q'c[ is bisected in 
 G, it follows that Qq' is parallel to PGP\ 
 
THE ELLIPSE, 99 
 
 Also, since CD is parallel to QQ and bisects Q'q\ GD 
 must also bisect Qq'. 
 
 Thus GD bisects the chord Qq' which is parallel to 
 PGP\ and it must therefore bisect every chord which is 
 parallel to PGP'. 
 
 Def. If two diameters of a conic are such that each 
 bisects all chords of the conic which are parallel to the 
 other, the two diameters are said to be conjugate to one 
 another. 
 
 69. Prop. XIII. The lines joining any point of an 
 ellipse to the two extremities of any diameter are parallel 
 to conjugate diameters. 
 
 Q 
 
 Let PGP' be any diameter of the ellipse, and Q be 
 any point on the curve. 
 
 Join QP, QP' and let Fbe the middle point of QP, 
 
 Then, since V is the middle point of PQ and G the 
 middle point of PP', GV is parallel to QP'. 
 
 Thus the diameter conjugate to PQ is parallel to QP', 
 as was to be proved. 
 
 Conversely, if P, Q, P' be three points on an ellipse, 
 such that QP, QP' are parallel to some pair of conjugate 
 diameters, then will PP' be a diameter. 
 
 Def. The straight lines joining any point on an 
 ellipse to the two extremities of any diameter, are called 
 supplemental chords. 
 
 7—2 
 
100 
 
 THE ELLIPSE. 
 
 70. Prop. XIV. If the sides of a parallelogram 
 touch an ellipse, its diagonals will he conjugate diameters 
 of the ellipse. 
 
 Let KLMN be any parallelogram whose sides touch 
 an ellipse, and let P, P' be the points of contact of one 
 pair of parallel tangents, and Q, Q' the points of contact 
 of the other pair. 
 
 Then, since the tangents at P, P' are parallel, PGP' 
 and similarly QCQ\ is a diameter, and therefore PQP'Q' 
 is a parallelogram, so that PQ and P'Q' are parallel. 
 
 Now CK bisects PQ, and therefore also the parallel 
 chord P'Q. Hence KCM is a straight line, and it is 
 clearly parallel to PQ' or QP'. 
 
 Similarly LCN is a straight line, and GL bisects the 
 chord PQ' which is parallel to KGM. 
 
 Hence KM and LN are conjugate diameters of the 
 ellipse. 
 
 Conversely, if any two conjugate diameters of an 
 ellipse are cut by any tangent in the points K, L, the other 
 tangents to the ellipse from K and L will he parallel. 
 
THE ELLIPSE. 101 
 
 71. Prop. XV. If the tangents at the extremities 
 of a chord QQ' of an ellipse meet in jP, and the diameter 
 CT cut QQ' in V and the curve in P ; then will 
 
 GV,GT=GP\ 
 
 We know [Art. 18] that the tangent at P is parallel 
 to QQ'. Let this tangent cut TQ in K. 
 
 Draw PL parallel to TQ meeting QQ' in L. 
 
 Then, since PKQL is a parallelogram, KL will bisect 
 PQ. 
 
 But we know [Art. 18] that KG will bisect PQ, whence 
 it follows that KLG is a straight line. 
 
 Then, since i F is parallel to KP, 
 
 GV:GP=GL: GK 
 And, since PL is parallel to TQ, 
 
 GL:GK==GP:GT. 
 
 Hence GV.GP^GP: GT, 
 
 or CV.GT=GP\* 
 
 Prop. III. and Prop. IV. are particular cases of the 
 above general theorem. 
 
 * This proof was first given by Dr C. Taylor. 
 
102 
 
 THE ELLIPSE. 
 
 72. Prop. XVI. If the tangent at any point P of an 
 ellipse whose foci are S, S' be cut by any pair of parallel 
 tangents in the points T, T' , and CI) be the semi-diameter 
 conjugate to OP ; then will 
 
 TP.Pr = SP.ST = GD\ 
 
 Let Q, Q' be the points of contact of the parallel 
 tangents. Join 8T, S'T, 8T\ S'T. Produce SP to H 
 making PH = PS\ and join HT, HT. 
 
 Then, since TT' bisects the angle HP 8' and PH = PS\ 
 it follows that TH=^T8' and HT^S'T and therefore 
 that the triangles HTT\ S'TT are equal in all respects. 
 
THE ELLIPSE. 103 
 
 Hence /.HTT^aT'TS' 
 
 = ZSTQ; [Prop. XL] 
 
 .-. ZlITS = zQTr. 
 
 Similarly Z HT'S = Z QTT. 
 
 Hence Z HTS + Z HT'S = Z QTT' + Z QTT 
 
 = 2 right angles, since TQ and 
 T'Q' are parallel. 
 
 Thus S, T, II, T' lie on a circle, and therefore 
 
 TP. PT = SP.PH= SP. ST. 
 
 Now let the tangent at P be cut in t, t' by the two 
 tangents parallel to GP ; then 
 
 SP.ST = tP.Plf 
 
 = CD', since tP = P^' = GD. 
 
 Thus rP. Pr = SP, ST = GD\ 
 
 Cor. Since the diagonals of a parallelogram whose 
 sides touch an ellipse are conjugate diameters, the propo- 
 sition may be enunciated thus : 
 
 If the tangent at P to an ellipse he cut in T, T hy any 
 pair of conjugate diameters ; then will 
 
 TPTT^ = SP.ST = GB\ 
 
 It should be noticed that since the angles TST' 
 and THT' are supplementary, the angles TST and TS'T 
 are supplementary; and therefore the portion of any 
 tangent to an ellipse intercepted between any pair of 
 parallel tangents, or between any pair of conjugate 
 diameters, subtends supplementary angles at ike foci. 
 
104 
 
 THE ELLIPSE. 
 
 73. Prop. XVII. The sum of the squares of conjugate 
 diameters of an ellipse is constant 
 
 Let CP, CD be any pair of conjugate semi-diameters 
 of an ellipse. Draw the tangents at P, D cutting the 
 axis major in T, K respectively ; and let PN, DM be 
 perpendicular to the axis major. 
 
 Then PT is parallel to CD and DK parallel to OP. 
 
 Produce NP, MD to meet the auxiliary circle in p 
 and d respectively. 
 
 Then, since ON . GT = GA^^ = Gp% it follows that pT 
 must be the tangent at p to the auxiliary circle. Simi- 
 larly Kd touches the auxiliary circle at d. 
 
 Since PT is parallel to GL, the right-angled triangles 
 TJSrP, GMD must be similar. 
 
 Hence 
 
 TN '.GM = NP : MD 
 
 = Np:Md, 
 
 whence it follows that the triangles TNp and GMd are 
 similar, and therefore that Gd is parallel to Tp. But Tp 
 touches the auxiliary circle and is therefore perpendicular 
 to Gp. 
 
THE ELLIPSE. 105 
 
 Hence Gp and Cd are at right angles, whence it 
 follows that GM = Np and GJSf = Md. 
 
 Now GP' + GD' = GN' + NP' + GM'' + MD\ 
 
 But GN' + GM' = (7iV^2 + Np' = 0^^ ; 
 
 and NP' + MD' : Fp' + Md' = BC' : GA'\ 
 
 hence, as JSTp' + Md' = Np' + GJST' = GA', 
 FP' + MD' = GB'. 
 Thus GP' + GD' = GA' + GB'. 
 
 Or thus 
 
 (>SfP + S'P)' = SP' + >SfT^ + 2SP . ^T. 
 
 But SP + 8T=^2GA, 
 
 SP . >SfT = GD', [Prop. XVL] 
 
 and SP' + /SfT^ = 2(7P2 + 2GS'. 
 
 Hence 4(7^^ = 2GP' + 2GS' + 2GD' ; 
 
 .-. GP' + (71)2 == 2(7^2 _ (7^2 = 0^2 + ((7^2 _ O/Sf^) 
 
 =^GA'+GB'. 
 
 EXAMPLES. 
 
 1. Shew that two diameters of an ellipse which make equal angles 
 with either axis are equal to one another. 
 
 2. Shew that the equal conjugate diameters of an elHpse are parallel 
 to the lines joining an extremity of one axis to the two extremities of the 
 other. 
 
 3. Shew that if a parallelogram he inscribed in an ellipse, the 
 diagonals of the parallelogram will be diameters of the ellipse. 
 
 4. Shew that, if CP, CD be conjugate semi-diameters of an ellipse, 
 and if the tangents at P, D meet in T, and CT cut the curve in R and PD 
 in F, then will 2CV^=CR^ and GT^=2CR'. Hence find the loci of V 
 and T for different pairs of conjugate diameters. 
 
 5. Shew that, if P, D be any two points on an ellipse, and p, d the 
 corresponding points on the auxiliary circle, and if the tangents at P, D 
 meet in T and the tangents at p, d meet in t ; then will Tt be perpen- 
 dicular to the major axis of the ellipse. Hence shew that, if pCd be a 
 constant angle, the locus of T will be an ellipse. 
 
106 
 
 THE ELLIPSE. 
 
 74. Prop. XVIII, The parallelogram formed hy the 
 tangents at the ends of conjugate diameters of an ellipse, is 
 of constant area. 
 
 Let the tangents at the extremities of the conjugate 
 diameters PGP\ BCD' form the parallelogram KLK'U, 
 Let the tangent at P cut the major axis in T, and draw 
 PN, DM perpendicular to the axis. 
 
 Then rKK' = 4t\\^KG = 8 A BGP 
 
 =SADCT 
 =^4DM.GT. 
 Now it can be proved as in Prop. xvii. that 
 J)M:GN'=BG:AG; 
 .'. BM,GT'.GN.GT=BG.AG:AG\ 
 But GN.GT= GA' and therefore BM.GT = BG.AG. 
 Thus the parallelogram formed hy the tangents at the 
 ends of any pair of conjugate diameters of the ellipse is 
 constant and equal to 4fAG.BG. 
 
 Cor. If the normal at P cut the conjugate diameter 
 in the point F, then PF.GB = AG.BG. 
 
THE ELLIPSE. 
 
 107 
 
 75. Prop. XIX. If the normal at any point P of 
 an ellipse cut the major and minor axes respectively in 
 the points G, g, and the diameter conjugate to OP in t^ie 
 point F; then PF. PG=BG' and PF.Pg = AG\ 
 
 Let the tangent at P cut the major and minor axes 
 respectively in the points T, t 
 
 Draw PN, Pn perpendicular to the axes, and produce 
 to meet the conjugate diameter in the points K, k respec- 
 tively. 
 
 Then a circle will circumscribe GFKN^ since the angles 
 at N and F are right angles ; 
 .-. PF.PG = PN.PK 
 
 = Gn. Gt, since NP = Gn and KP = Gt 
 = BG\ 
 Again, a circle will circumscribe gFnk, since the angles 
 gFk and gnk are right angles ; 
 .-. PF.Pg = Pn.Pk 
 
 = GN . GT, since GIST = nP and GT = kP 
 = GA^ 
 
108 
 
 THE ELLIPSE. 
 
 Since PF . GI) = AC . BG, [Prop, xviil] 
 
 it follows from the above that 
 
 PG:GI) = BG:AG, 
 Pg:GD = AG:BG, 
 and PO.Pg^GD\ 
 
 76. Let SP cut CD in the point E ; then we know that PE = CA. 
 Hence PE^=PF .Pg^ whence it follows that the angle PEg is 
 a right angle. 
 
 Also, if GL be perpendicular to SP, the points L, G, F, E are on a 
 circle, and therefore PL.PE = PF.PG, that is PL.AC=B(P, so 
 that PL is equal to the semi-latus-rectum. 
 
 Thus tbe projections of PG and Pg on tlie focal distance SP 
 are respectively equal to the semi-latus-rectum and the semi- 
 major axis. 
 
 77. When a pair of conjugate diameters of an ellipse 
 are given, the axes, foci, &c. can be determined. 
 
 For let PGP', DGD' be the two given conjugate dia- 
 meters, and let the normal at P cut DGD' in F. 
 
 Then, if G, g be the unknown points in which the 
 normal cuts the major and minor axes respectively, and if 
 be the middle point of Gg, we know that 
 
 PF.PG=^BG^ 
 
 and PF .Pg^AG\ 
 
 and therefore PF{PG-\- Pg) ^AG^ + BG\ 
 
THE ELLIPSE. _^ _^^^^ 
 
 Hence 2Pi^ .PO=^GP' + GD\ fflT]^> ^^ ^ 
 
 from which FO can be found. \^ ^J^ 
 
 Then, having found PO from the relation ^J^^^^ o^ * 
 2PF . PO = CP^ + CD\ ^^f'^nv ^^ 
 
 draw a circle with as centre and 00 sls radius, and this 
 circle will cut the normal at P in the points G, g on the 
 axes. Thus the directions of the axes are found, and the 
 lengths of the semi-axes are given by the relations 
 
 PF .PG^BG\ 
 and PF.Pg =^ AG\ 
 
 Having found the axes of the ellipse, the foci and 
 directrices can be easily found. 
 
 EXAMPLES. 
 
 1. Shew that the area of a parallelogram inscribed in an ellipse is 
 greatest when its diagonals are conjugate diameters. 
 
 2. Shew that the area of a parallelogram whose sides touch an 
 ellipse cannot be less than that of the parallelogram formed by the 
 tangents at the ends of the axes. 
 
 78. We have already proved in Art. 24 that the ratio 
 of the rectangles contained by the segments of any two 
 intersecting chords of an ellipse, which are parallel to two 
 given straight lines respectively, is constant for all posi- 
 tions of the point of intersection of the chords ; and, by 
 considering the parallel chords through the centre of the 
 ellipse, it follows that the ratio of the rectangles contained 
 by the segments of any two chords of an ellipse is equal to 
 the ratio of the squares of the parallel semi-diameters ; and, 
 as a particular case, the lengths of the two tangents drawn 
 from any point to an ellipse are in the ratio of the parallel 
 semi-diameters. 
 
 79. Prop. XX. The length of a focal chord of an 
 ellipse varies as the square of the parallel semi-diameter. 
 
 Let PBF be the focal chord and DGD' the parallel 
 diameter. 
 
110 THE ELLIPSE. 
 
 Let the tangents at P, P' meet in T\ then GT will 
 bisect PP' in F, and GVT will^be parallel to the tangent 
 atD. 
 
 Let the tangent at P cut CD produced in E, and let 
 PTT be drawn parallel to CV meeting CD in W. 
 
 Then GW . GE = GD\ [Prop, xv.] 
 
 But GW=YP = \PF\ 
 
 and GE = AG, [Prop, vn.] 
 
 Hence PP'.GA = 2GD\ 
 
 80. Prop. XXI. If a circle cut an ellipse in four 
 points, the line joining any two of the points and the line 
 joining the other two points make equal angles with either 
 axis of the ellipse. 
 
 For let K, L, M, iV be the four points of intersection. 
 
 Let KL and MJS^ intersect in 0. Then, since the 
 four points K, L, M, N are on the circle, the rectangles 
 KO . OL and MO . ON are equal. 
 
 Also, since the four points are on the ellipse, 
 
 KO.OL : MO,ON^GP^ : GP'\ 
 
 where GP, GP' are the semi-diameters parallel to KL and 
 MN respectively [Art. 78]. 
 
 Hence the semi-diameters parallel to the chords are 
 equal, and they must therefore be equally inclined to 
 either axis. 
 
 Secondly, let the chords KL and MN be parallel. 
 
THE ELLIPSE. Ill 
 
 Then, since the line joining the middle points of any 
 two parallel chords of a circle is perpendicular to the 
 chords, it follows that the chords KL and MN are perpen- 
 dicular to the diameter of the ellipse which is conjugate 
 to them, and therefore the chords must be parallel to one 
 of the axes. 
 
 Conversely, if two chords of an ellipse, which are not 
 parallel, be equally inclined to an axis, their four ex- 
 tremities will lie on a circle. 
 
 EXAMPLES. 
 
 1. If OP, OQ be two tangents to an ellipse whose semi-axes are 
 CA, CB, OP : OQ cannot be greater than CA : GB. 
 
 2. Shew that a circle cannot cut an ellipse in more than four 
 points. 
 
 3. If the chords PQ, PQ' of an ellipse be equally inclined to an axis, 
 the circle PQQ' will touch the ellipse at P. 
 
 81. Def. A line QV, drawn from any point Q of an 
 ellipse parallel to the tangent at either extremity of the 
 diameter PVGP\ is called an ordinate to the diameter 
 POP'. 
 
 Prop. XXII. If QY he the ordinate to any dia- 
 meter P YCP' of an ellipse, and CD he the semi-diameter 
 conjugate to PGP' ; then will 
 
 QY : CP' - GY' = CD' : GP^ 
 
112 
 
 THE ELLIPSE. 
 
 Let QV produced cut the ellipse in Q' ; then, since QQ' 
 is parallel to the tangent at F, V will be the middle 
 point of QQ'. 
 
 Now we know [Art. 78] that the rectangles contained 
 by the segments of two chords of an ellipse drawn in 
 given directions is constant. Hence 
 
 QV. Vq : PV. yP' = BG . CD' : PC. GP' ; 
 
 .-. QV'iPV.VP'^nC'.PG^ 
 
 or QV : GP' - GV == GD' : GP'. 
 
 82. The following generalizations of some of the pre- 
 ceding theorems are interesting. 
 
 M 
 
 \ \\>P.-^?^^"^ 
 
 t 
 
 \ ot^ 
 
 ?^.^ 
 
 n \ ^^ 
 
 /TaI 1 
 
 
 ^^^' 1^' 
 
 
 
 
 w 
 
 X' 
 
 Let QQ' be any chord of an ellipse. Draw SY perpendicular to 
 the chord and let SY produced cut the directrix in Z. 
 
 Then we know that CZ will bisect the chord and that the tangents 
 at Q, Q' will meet at some point P on GZ. 
 
 Draw the perpendiculars PM, PN on the directrix and transverse 
 axis respectively. Draw through P a line perpendicular to QQ' 
 meeting QQ' in W, the major axis in G and the minor axis in g. Let 
 QQ' cut the axes in T, t. 
 
THE ELLIPSE. 113 
 
 Then, since PG is parallel to SZ, 
 
 GG :CS=GP :CZ 
 
 = GN : GX, since FN and ZX are parallel. 
 
 Hence CG : CIV=CS : CX (i). 
 
 Also SG : SG=ZP : ZC 
 
 =PM: CX; 
 
 .: SG :FIIII = SC : CX (ii). 
 
 Again, since SG : PM=SG : GX 
 
 = S'G\M'P, 
 
 it follows that PG goes through the intersection of MS and M'S' ; 
 
 /. Gn:gG=XM :gG; 
 
 :. Cn : gC = XS : 8C (iii). 
 
 Also, since PG : Pg=MS : Mg ; 
 
 .-. PG : Fg=XS : XC = BC2 : CA^ (iv). 
 
 Now, since TK is perpendicular to SZ, and ZK perpendicular to SZ, 
 it follows that SK is perpendicular to ZT. 
 
 But we know that SK is perpendicular to SP. 
 
 Hence ZT is parallel to SF (v). 
 
 Then GN : GX=GP : GZ 
 
 = OS : Cr, since ZT is parallel to SP ; 
 
 .'. Clf .CT = CS.CX=CA2 (vi). 
 
 Also GG :GS=GP :GZ 
 
 = GS : GT, from (v) ; 
 
 .-. CG. CT=CS2 (vii). 
 
 Again, since PGg is perpendicular to Tt, and TG perpendicular 
 to tg, it follows that G is the orthocentre of the triangle Ttg, and 
 therefore that 
 
 tC.Cg=CG.CT=CS2 (viii). 
 
 And, since tG .Gg = SG .GS', it follows that S, t, S', g are on a 
 circle; and tg must be the diameter of this circle, since tg bisects 
 SS' at right angles, whence it follows that W is also on the circle. 
 
 Thus S, S', t, g and W are a circle (ix). 
 
 From (iii) and (viii) it follows that 
 
 Cn . Ct=CB2 (x). 
 
 Moreover, if a line through G parallel to the chord OQ' cut PGg 
 in F, it can be easily proved, as in Art. 75, that 
 
 FP . FG = BC2, and FP . Fg=:AC2 (xi). 
 
 When P is on the ellipse the relations (i), (ii), (iii), &c. reduce 
 to those proved in Articles 57, 58, 59, &c. 
 
 S. C. 8 
 
114 EXAMPLES ON THE ELLIPSE. 
 
 EXAMPLES ON THE ELLIPSE. 
 
 1. Tangents are drawn to an ellipse from any point on the line 
 through a focus perpendicular to the axis. Shew that the length 
 intercepted by the tangents on the corresponding directrix is bisected 
 by the axis. 
 
 2. A semi-circular piece of paper is folded over so that a particular 
 point K of the bounding diameter lies somewhere on the circular 
 boundary: prove that the line of the crease touches an ellipse whose 
 foci are £" and C, G being the centre of the circle. 
 
 3. From any point on the auxiliary circle of an ellipse tangents are 
 drawn touching the ellipse in P, Q, and POP', QGQ' are diameters of 
 the ellipse. Shew that PQ' and P'Q are focal chords. 
 
 4. Two tangents TP, TQ are drawn from any point T to an ellipse, 
 and any straight line parallel to TP meets TQ in L, PQ in M, and the 
 conic in B, B\ Shew that LM^=LR . LR'. 
 
 5. The tangent at a point P of an ellipse whose foci are S, S' cuts 
 the tangents at the vertices in the points T, T', and TS, T'S' meet in Q. 
 Shew that Q is on the normal at P and that the circle SPS' is the nine 
 point circle of the triangle QTT'. 
 
 6. From the foci S, S' of an ellipse lines are drawn perpendicular 
 to SP, S'P respectively, and meeting the normal at P in O, 0' respectively. 
 Shew that 00' is bisected by the minor-axis. 
 
 7. PSQ, P'S'Q' are parallel focal chords of an ellipse ; shew that 
 the tangents at P, Q, P', Q' form a parallelogram two of whose angular 
 points are on the directrix and the remaining two on the auxiliary 
 circle. 
 
 8. Shew that, if P, D be extremities of a pair of conjugate diameters 
 of an ellipse, the lines joining P and D to the foci will touch a circle 
 whose radius is equal to the semi-minor axis. 
 
 9. P, p are corresponding points on an ellipse and its auxiliary circle. 
 Shew that the perpendicular distances of the foci S, S' from the tangent 
 to the circle at p are equal to /SP, S'P respectively. 
 
 10. The sides of the parallelogram ABCD touch an ellipse whose 
 focus is S; shew that the circles ABS, BGS, GDS, DAS are all equal. 
 
 11. If a focal chord of a conic pass through the extremities of a 
 pair of conjugate diameters of an ellipse, the length of the chord will be 
 equal to the semi-axis-major. 
 
 12. A line AB is drawn from a fixed point A to meet a fixed circle 
 in B; through B a line BG is drawn perpendicular to AB to meet a 
 concentric circle in G. Shew that a line through G parallel to AB will 
 touch a fixed conic. 
 
EXAMPLES ON THE ELLIPSE. 115 
 
 13. Any line through the focus of an ellipse cuts the tangents at the 
 vertices in the points T, T' respectively. Shew that the circle whose 
 diameter is TT' touches the ellipse at the ends of a chord parallel to the 
 major axis. 
 
 14. If two ellipses have the same auxiliary circle and one pass 
 through the foci of the other, the second will pass through the foci of 
 the first. 
 
 15. AA' is the major axis of an ellipse of which S, S' are the foci 
 and P any point on the curve; AE, A'E' are drawn parallel to /SP, S'P 
 respectively to meet the tangent at P in B, R'. Shew that 
 
 AE + A'R'=AA'. 
 
 16. A parabola is described passing through the foci of a given 
 ellipse and having for focus any point on the ellipse; shew that the 
 directrix of the parabola always touches the auxiliary circle of the ellipse. 
 
 17. The tangents at P, D the extremities of conjugate diameters of 
 an ellipse whose centre is G meet in T, and GT cuts the ellipse in Q, and 
 the chords QE, QE' are drawn parallel to OP, GD respectively. Shew 
 that RE' is parallel to PD. 
 
 18. Lengths GA, GB are taken on any two given straight lines so 
 that the sum of the squares of GA and GB is equal to a given square, and 
 the parallelogram AGBP is completed. Shew that the locus of P is an 
 ellipse. 
 
 19. OF is the perpendicular from the centre on the tangent at any 
 point P of an ellipse and Q is the point of contact of the other tangent 
 from Y. Shew that the normal at P passes through the other extremity 
 of the diameter through Q. 
 
 20. Perpendiculars SY, S'Y' are drawn from the foci S, S' upon a 
 pair of tangents TY, T'Y' to an ellipse; prove that the angles STY, S'TY' 
 are equal or complementary to the angles at the base of the triangle 
 GYY'. 
 
 21. If the ordinate at a point P of an ellipse meet the auxiliary 
 circle in^, shew that the diameter of the circle through p and the normal 
 at P to the ellipse meet on a fixed circle. 
 
 22. Any point P of an ellipse is joined to A, A\ the extremities of 
 the major axis, and AF is drawn perpendicular to A'P. AP and AF 
 meet the tangent at A' in K and L. Prove that A'K : A'L = BC^ : AG'^. 
 
 23. Through any point P of an ellipse a line is drawn perpendicular 
 to the diameter GP meeting the auxiliary circle in R, E'. Shew that 
 RR' is equal to the differences of the focal distances of an extremity of 
 the diameter conjugate to GP. 
 
 24. Through a point P of an ellipse a line PDE is drawn cutting the 
 axes so that the segments PD and PE are equal to the two semi-axes 
 respectively : perpendiculars to the axes through E and D intersect in 0. 
 Prove that O is a normal. 
 
 8—2 
 
116 EXAMPLES ON THE ELLIPSE. 
 
 25. -P is any point on an ellipse whose centre is G and whose foci 
 are S, S'. The diameter conjugate to CP cuts SP in E. Shew that the 
 difference of the squares of GP and SE is constant. 
 
 26. An ellipse of given semi-axes touches three of the sides of a given 
 rectangle ; find its centre and focus. 
 
 27. If a chord of an ellipse be parallel to one of the equi-con jugate 
 diameters, the normals at its extremities will meet on the diameter of 
 the ellipse which is perpendicular to the other equi-conjugate diameter. 
 
 28. If C'-Pj GD are conjugate semi-diameters of an ellipse, and the 
 normals at P, D meet in O ; then will GO be perpendicular to PD. 
 
 29. The normals at P, P', the extremities of a focal chord of an 
 ellipse, cut the axis major in G, G' respectively and intersect in the point 
 O. Shew that the line through O parallel to PP' will bisect GG'. 
 
 30. Two given ellipses in the same plane have a common focus and 
 one revolves about the common focus while the other remains fixed: 
 prove that the locus of the point of intersection of their common tangents 
 is a circle. 
 
 31. A focus S of an ellipse, the length of the major axis, and one 
 point P on the curve are given. Shew that the ellipse always touches a 
 fixed ellipse whose foci are S and P. 
 
 32. Having given the focus of an ellipse, the length of the major 
 axis, and that the second focus lies on a fixed straight line, prove that 
 the ellipse will touch two fixed parabolas having the given focus for 
 focus. 
 
 33. The centre of an ellipse, the radius of its director-circle, and one 
 point on the curve are given. Shew that the ellipse always touches a 
 fixed concentric ellipse. 
 
 34. Two given concentric ellipses in the same plane have equal 
 major axes. Shew how to draw their common tangents. 
 
 35. Shew that, if two of the diagonals of a quadrilateral circumscrib- 
 ing an ellipse intersect in a focus, these diagonals must be at right angles 
 and the third diagonal must be the corresponding directrix. 
 
 36. If PP' be a chord of an ellipse parallel to the major axis, and 
 the two circles be drawn through a focus S touching the conic at P, P' 
 respectively ; prove that the circles will intersect at a point F which is the 
 intersection of PP' and ST, where T is the point of intersection of the 
 tangents at P, P'. 
 
 Prove also that the locus of P, for different positions of PP', will be a 
 parabola whose vertex is S. 
 
 37. A circle cuts off equal chords AB, CD from two given parallel 
 straight lines, and passes through a fixed point S between the hnes ; prove 
 that the intersecting straight lines AD, BG always touch a fixed conic of 
 which S is one focus. 
 
EXAMPLES ON THE ELLIPSE. 117 
 
 38. If -Pj P be corresponding points on an ellipse and the auxiliary 
 circle, centre G, and if GP be produced to meet the auxiliary circle in q, 
 prove that the tangent at the point Q on the ellipse corresponding to q is 
 perpendicular to Gp, and that it cuts off from Gp a length equal to GP. 
 
 39. Two ellipses lie in the same plane and have one common focus 
 and equal major axes ; one ellipse revolves in its own plane about the 
 common focus, the other ellipse remaining fixed. Prove that the com- 
 mon chord of the two ellipses always touches another ellipse confocal 
 with the fixed ellipse. 
 
 40. The tangent at any point P of an ellipse meets any pair of 
 parallel tangents in M, N. Prove that the circle described on MN as 
 diameter will meet the normal at P at points whose distance apart is 
 equal to the diameter conjugate to GP, and whose distances from the 
 centre are respectively equal to the sum and the difference of the semi- 
 axes. 
 
 41. The diameter parallel to any focal chord of an ellipse is equal to 
 the chord joining the points on the auxiliary circle corresponding to the 
 extremities of the focal chord. 
 
 42. The sides of a rectangle touch an ellipse ; shew that a circle 
 through a focus and any two adjacent angular points of the rectangle is 
 equal to the auxiliary circle. 
 
 43. A.ny tangent to an ellipse cuts the director-circle in the points 
 P, Q and a directrix in the point K. Shew that, if S be the corresponding 
 focus, the triangles KSP, SKQ will be similar. 
 
 44. The area of the parallelogram formed by the tangents at the 
 extremities of any two diameters of an ellipse varies inversely as the 
 area of the parallelogram formed by joining the points of contact. 
 
 45. A system of parallelograms is inscribed in an ellipse whose sides 
 are parallel to the equi-conjugate diameters : prove that the sum of the 
 squares of the sides of any parallelogram is constant. 
 
 46. An ellipse is described having the line joining the foci of a given 
 ellipse for a diameter and for that conjugate to it a line equal to the 
 minor axis of the given ellipse : prove that it always touches the 
 original ellipse. 
 
 47. A series of ellipses have their foci on two adjacent sides of a 
 given parallelogram and touch the other two sides. Shew that the locus 
 of their centres is a straight line, 
 
 48. -^n extremity of a diameter of a central conic is joined to the 
 two extremities of one of its ordinates : prove that the chords so drawn 
 are proportional to the diameters parallel to them. 
 
 49. Normal chords of a conic which are at right angles are propor- 
 tional to the parallel diameters. 
 
 50. If PP' he a chord of an ellipse normal at P, GY the perpendicular 
 on the tangent at P, and GQ the semi-diameter parallel to PP'; then 
 willPP'.Cr=2(7Q2. 
 
118 EXAMPLES ON THE ELLIPSE. 
 
 51. POP' is any diameter of an ellipse, PQ is any chord which is 
 produced so as to cut the tangent at P' in JR. Shew that the diameter 
 parallel to PQ is a mean proportional between PQ and PR. 
 
 52. From any point on an ellipse two chords are drawn equally 
 inclined to the tangent at the point. Shew that these chords are to one 
 another as the parallel focal chords. 
 
 53. An ellipse of given major axis has one focus at the focus of a 
 given parabola and touches the parabola; shew that the locus of the 
 centre of the ellipse is a straight line perpendicular to the axis of the 
 parabola. Shew also that all the ellipses touch another parabola co-axal 
 and confocal with the given one. 
 
 54. The tangent at any point P of an ellipse cuts the directrices 
 corresponding to the foci S, S' in the points Z, Z' respectively. Shew 
 that ZS, Z'S' will meet on the ordinate through the other extremity of 
 the diameter through P. 
 
 55. The line joining the foci of an ellipse subtends at the pole of 
 any chord an angle equal to half the sum of the angles which it subtends 
 at the extremities of the chord. 
 
 56. The ordinate at any point P of an ellipse is produced to meet 
 the perpendicular from the centre on the tangent at P in Q. Shew that 
 the locus of Q is an ellipse. 
 
 57. -P-P' is a diameter of an ellipse and Q is a point such that the 
 tangents at P and Q are at right angles. Shew that PQ, P'Q make equal 
 angles with the tangent at Q. Prove also that if R be any other point 
 on the ellipse QP + QP' is greater than RP + RP'. 
 
 58. Shew that the perimeter of a parallelogram inscribed in an 
 ellipse cannot be greater than that of the parallelogram whose diagonals 
 are the axes of the ellipse. 
 
 59. PP' is any chord of an ellipse and PG, P'G' are the normals at 
 P, P', the points G, G' being on the major-axis. Shew that the 
 projections of PG, P'G' on the chord PP" are equal. 
 
 60. Construct an ellipse having given the centre, a tangent, the 
 length of the major-axis, and a point through which a directrix passes. 
 
 61. The chord of contact of the tangents drawn from a point to 
 an ellipse cuts the major axis in T, and the line through O perpendicular 
 to the chord cuts the major axis in G. Shew that the circle whose dia- 
 meter is TG is cut orthogonally by a fixed circle. 
 
 62. Shew that, if any point P on an ellipse be joined to the foci and 
 the joining lines be produced to meet the corresponding directrices in 
 R, R' ; then RR' and the tangent at the other extremity of the diameter 
 through P will meet on the axis of the ellipse. 
 
 63. SY, S'Y' are the perpendiculars from the foci on the tangent at 
 any point P of an ellipse ; X, X' are the feet of the directrices correspond- 
 ing to the foci S, S' respectively. Shew that XY, X'Y' will meet on the 
 minor axis, and that NY, NY' will be perpendicular to XY, X'Y\ N being 
 the foot of the ordinate through P. 
 
EXAMPLES ON THE ELLIPSE. 119 
 
 64. If ^^' is the transverse axis of an ellipse, and if Y, Y' are the 
 feet of the perpendiculars let fall from the foci on the tangent at any 
 point of the curve, prove that the locus of the point of intersection of 
 ^r and A'Y' is an ellipse. 
 
 65. Any two conjugate diameters of an ellipse cut the director 
 circle in the points R, R' ; shew that RR' touches the ellipse. 
 
 66. Shew that, if SY, S'Y' be the perpendiculars from the foci on the 
 tangent at any point P of an ellipse, the tangents to the auxiliary circle 
 at F, Y' will meet in a point T on the ordinate NP, and the locus of T 
 will be an ellipse. 
 
 67. 'S'F, S'Y' are the perpendiculars from the foci S, S' on any 
 tangent to an ellipse, and X, X' are the feet of the corresponding 
 directrices ; XY, X'Y' cut the auxiliary circle again in Z, Z' respectively. 
 Shew that ZZ' will touch the ellipse. 
 
 68. The area of a triangle whose vertices are any three points on a 
 given ellipse is in a constant ratio to the area of the triangle whose 
 vertices are the three corresponding points on the auxiliary circle. 
 
 69. Shew that if a triangle inscribed in an ellipse be of maximum 
 area, its centroid will coincide with the centre of the ellipse. 
 
 70. Having given one focus of an ellipse, the directrix corresponding 
 to the other focus, and also a tangent to the ellipse. Find the centre. 
 
 71. A system of ellipses is drawn having a common focus S and a 
 common latus-rectum LSL'. A fixed straight line through S is drawn 
 to intersect the conies and the normals are drawn at the points of 
 intersection. Shew that these normals all touch a parabola whose focus 
 lies on LSL'. 
 
 72. The minor-axis of an ellipse inscribed in a given triangle 
 cannot exceed the diameter of the inscribed circle. 
 
 73. An ellipse touches the sides of a given triangle and has a given 
 centre. Find the points of contact. 
 
 74. If PSQ be a focal chord of an ellipse, and Pp, Qq be chords 
 perpendicular to it, then the triangles SPp, SQq will be similar. 
 
 75. Shew that the exterior angle between two tangents to an ellipse 
 is equal to half the sum of the angles the chord of contact subtends at 
 the foci. 
 
 76. If the double ordinate PP' drawn perpendicular to the major 
 axis of an ellipse whose centre is G meet the auxiliary circle in p and p', 
 shew that the part of the normal at P intercepted between Gp and Cp' is 
 bisected at P. 
 
 77. The normal to an ellipse at P cuts the axes in G, g, and GK is 
 the central perpendicular on the tangent at P. Shew that, if 0, 0' be 
 the middle points of GG, Gg respectively, then OB = OK=OP and 
 0'A = 0'K=0'P. 
 
120 EXAMPLES ON THE ELLIPSE. 
 
 78. -4 is a fixed point within a given circle, P is any point on the 
 circumference of the circle and PQ is drawn making a given angle with 
 AP. Shew that PQ touches a fixed ellipse, and that the eccentricity of 
 the ellipse is independent of the value of the fixed angle APQ. 
 
 79. PNP' is any double ordinate of an ellipse whose centre is C, and 
 the normal at P cuts GP' in Q. Shew that the locus of Q, for different 
 positions of PNP', is an ellipse. 
 
 80. If a pair of tangents to an ellipse be at right angles to one 
 another, the rectangle contained by the perpendiculars from the centre 
 and the intersection of the tangents on the chord of contact is constant. 
 
 81. A parallelogram circumscribes a given circle, and two of the 
 angular points are on fixed straight lines parallel to one another and 
 equidistant from the centre ; shew that the other two are on an ellipse of 
 which the given circle is the minor auxihary circle. 
 
 82. From the centre G of two concentric circles two radii GQ, Gq 
 are drawn equally inclined to a fixed straight hne, the first to the outer 
 circle, the second to the inner: prove that the locus of the middle point 
 P of Qq is an ellipse, that PQ is the normal at P to this ellipse, and that 
 Qq is equal to the diameter conjugate to GP. 
 
 83. From P any point on an ellipse a tangent is drawn to the 
 minor auxiliary circle cutting the director circle in Q, B; shew that PQy 
 QR are equal to the focal distances of P. 
 
 84. From a point T the tangents TP, TQ are drawn to an ellipse. 
 Shew that, if the bisector of the angle PTQ passes through a fixed 
 point on the major axis of the ellipse, T must lie on a fixed circle. 
 
 85. Shew that, if a circle roll on the inside of a circle of double 
 the radius, any point carried with the moving circle will describe an 
 ellipse. 
 
 86. Shew that, if P be any point on an ellipse whose foci are 8, S\ 
 the locus of the centre of the circle inscribed in the triangle SPS' is an 
 ellipse. 
 
 87. The portion of any normal chord of an ellipse intercepted 
 between the directrices subtends at the pole of the chord an angle equal 
 to half the sum of the angles subtended at the extremities of the chord 
 by the distance between the foci. 
 
 88. TQ, TQ' are any two tangents to an ellipse, and TN is the 
 perpendicular on the axis major. Shew that TN bisects the angle QNQ'. 
 
 89. The tangent at a fixed point P of an ellipse whose foci are S, S' 
 is cut by any pair of parallel tangents in the points T, T. The lines 
 TS, T'S' intersect in 0, and the lines TS\ T'8 in O' ; shew that O, 0' lie 
 on a fixed circle through the foci. 
 
 90. Three conies are inscribed in an acute-angled triangle, and the 
 three points of bisection of the perpendiculars from the angles on the 
 sides of the triangle are each a focus of one conic ; prove that these 
 three points form with the other three foci a hexagon of which the 
 opposite sides meet in the angles of the triangle. 
 
EXAMPLES ON THE ELLIPSE. 121 
 
 91. PG is the normal at any point P of an ellipse and OL is drawn 
 perpendicular to CP, and CM is drawn parallel to one of the focal 
 distances of P meeting PG in M. Shew that the triangles GLM and 
 MGP are similar. 
 
 92. Given a tangent to an ellipse, its point of contact, and the 
 director-circle, construct the ellipse. 
 
 93. Shew that, if two chords QQ\ BR' of an ellipse be drawn 
 parallel to one of the equi-conjugate diameters, meeting the other 
 equi-conjugate diameter PGP' in the points F, W on opposite sides of 
 the centre G such that IWG .GV=GP^j then will the normals at the 
 points Q, Q\ B, B' meet in a point on the diameter perpendicular to 
 PGP'. 
 
 94. An ellipse is inscribed in a triangle and having its centre at the 
 centre of the circum-circle of the triangle. Prove that the perpendiculars 
 from the corners of the triangle on the opposite sides will be normals to 
 the ellipse. 
 
 95. With the orthocentre of a triangle as centre two conies are 
 described, one touching the sides and the other passing through the 
 angular points ; shew that these conies are similar, their corresponding 
 axes being perpendicular. 
 
 96. PSQ, PHB are focal chords of an ellipse; prove that the 
 tangent at P and the chord QB cut the major axis in points equidistant 
 from the centre. 
 
 97. PSQ, PHB are focal chords of an ellipse, and the tangents at 
 Q, B meet in T. Shew that the middle point of TP is on the minor axis 
 and that the locus of T is an ellipse. 
 
 98. Two ellipses whose major axes are perpendicular intersect in 
 four points; shew that these four points lie on a circle. 
 
 99. Two ellipses whose major axes are parallel intersect in four 
 points ; shew that the four points will lie on a circle, unless the ellipses 
 have the same eccentricity. 
 
 100. If OP, OQ are two tangents to an ellipse and GP', GQ' be the 
 parallel semi-diameters, shew that OP . OQ + CP' . GQ'=OS . OH. 
 
CHAPTER IV. 
 The Hyperbola. 
 
 83. From the definition of a hyperbola as the locus 
 of a point which moves in the plane containing a given 
 point, called the focus, and a given straight line, called 
 the directrix, in such a manner that its distance from the 
 focus is always in a constant ratio greater than unity to its 
 perpendicular distance from the directrix, we have already 
 proved in Chapter I. that a hyperbola is symmetrical 
 about the line through its focus perpendicular to its 
 directrix ; and also that, if the line cut the curve in the 
 points A, A' , and G be the middle point of AA\ the 
 hyperbola is symmetrical about the line BOB' drawn 
 through C parallel to the directrix, whence it follows 
 that there is a second focus in the line ^AA' and a 
 second directrix perpendicular to A A'. 
 
 We have also proved in Art. 4 that, if B, & be the two 
 foci, and BAA'B' cut the directrices in the points X, X' 
 respectively, then 
 
 O/Sf : (7^ = 0^ : GX = BA : AX, 
 
THE HYPERBOLA. 
 
 123 
 
 The curve consists of two separate branches, as in the 
 figure, no part of the curve lying between the lines drawn 
 through the vertices parallel to the directrices. [Art. 6.] 
 
 The lines AG A' and BOB about each of which the 
 hyperbola is symmetrical are called respectively its trans- 
 verse and conjugate axes. 
 
 The axis AG A' cuts the curve in two points and the 
 intercepted portion A A' is also called the transverse axis. 
 
 X' 
 
 X 
 
 B'-- 
 
 The line BGB' does not cut the curve in real points ; 
 but if points B, F be taken on BGB' such that B'G=GB 
 
 and -BG'^ = GA^-G8^) 
 
 then BG^ = GS' - GA^ = (GS + GA)(GS-GA) = A'S.AS. 
 
 Also BG' = GS' - GA-" = GS'-GS. GX==GS.XS. 
 
 If LSL' be the latus-rectum, we have 
 
 SL:XS=^SA:AX 
 
 = GS : GA. 
 
 Hence SL . CA = CS .XS = BC. 
 
124 THE HYPERBOLA. 
 
 84. Prop. I. The difference of the focal distances of 
 any point on a hyperbola is constant. 
 
 Let 8y S' be the foci and P any point on the hyperbola. 
 Join 8P, S'P and draw PMM' perpendicular to and 
 meeting the directrices in the> points M, M\ 
 
 Then SP:MP=GA:GX, 
 
 and B'P:M'P=GA:GX 
 
 :. ST-SP:M'P-MP=GA:GX. 
 But WP -MP = M'M=X'X = 2CX. 
 
 Hence ST-8P = 2GA. 
 
 If P be any point on the branch through A, 
 8T-SP = 2GA; 
 and, if P be any point on the branch through A\ 
 SP-ST = 2GA. 
 
 It can easily be proved as in Art. 55 that, if Q be any point 
 external to the hyperbola, that is if Q be such that SQ is cut by the 
 curve in one and only one point between S and Q, then 
 
 Similarly, if Q be an internal point, that is if Q he such that 
 SQ is cut by the curve in two points or in none between S and 
 Q, then 
 
 SQ'-S'Q^AA'. 
 
THE HYPERBOLA. 
 
 125 
 
 The above property of a h3rperbola enables us to 
 describe the curve by the continuous motion of a point. 
 For, if the end J. of a rod AL be made to turn about the 
 fixed point A, and if one end of a string of constant length 
 be fastened to the rod at L and the other end be fastened 
 to a fixed point B^ and if the string be kept tight by a 
 
 pencil moving along the rod LA ; then, if P be any 
 position of the point of the pencil, LP + PB will be equal 
 to the length of the string, and LP + PA will be equal to 
 the length of the rod, so that ^P - BP will be equal to 
 the constant difference between the length of the string 
 and the length of the rod; and therefore P will always be 
 on a hyperbola whose foci are A and B. 
 
 EXAMPLES. 
 
 1. Construct a conic having given a focus and the two extremities 
 of its transverse axis. 
 
 2. Find the locus of the centre of a circle (1) which touches a given 
 line and a given circle, (2) which passes through a given point and 
 touches a given circle, and (3) which touches two given circles. 
 
 3. Find the locus of the centre of a circle which cuts two given 
 circles so that the common chords may each be equal to a given straight 
 line. 
 
 4. On a plane field the crack of the rifle and the thud of the baU 
 striking the target are heard at the same instant : find the locus of the 
 hearer. 
 
 5. The centre of a hyperbola, the length of the transverse axis, and 
 one point on the curve are given. Shew that the locus of the foci is 
 another hyperbola. 
 
126 
 
 THE HYPERBOLA. 
 
 85. Prop. II. The tcmgent at any point of a hyper- 
 bola is equally inclined to the focal distances of the point. ' 
 
 Let the tangent at P cut the directrices corresponding 
 to S, S' in K, K', respectively. 
 
 Draw PMM' perpendicular to the directrices. Join 
 8P, ST, SK, S'K'. 
 
 Then, from the similar triangles MPK, M'PK\ 
 KP:KT = MP:MT 
 = SP:ST. 
 
 Also we know [Art. 12] that the angles KSP, K'ST 
 are right angles. 
 
 Hence the triangles KPS, K'PS' are similar, and 
 we have 
 
 ZSPK=^ ZK'PS\ 
 
 Thus the tangent at P bisects the angle SPS\ 
 
 Since the normal is perpendicular to the tangent, it 
 
 follows that the normal bisects the angle between SP and 
 
 BT produced. 
 
THE HYPEKBOLA. 
 
 127 
 
 If the tangent at P cut the transverse axis in T, PT 
 bisects the angle SPB' and therefore 8'T : TS = ST : PS. 
 Now if P be any point on the branch through the vertex 
 A, SP will be greater than PS, and therefore ST will be 
 greater than TS. T must therefore lie between G and A. 
 
 86. Prop. III. If the tangent at any poirit P of a 
 hyperbola cut the transverse axis in T, and PN be the 
 perpendicular on the axis; then will ON . CT= GA\ 
 
 i.e. 
 
 Draw PMM' perpendicular to the directrices. 
 Then, since PT bisects the angle SPS\ 
 
 S'T: TS = SP:SP 
 
 =^M'P:MP = X']Sf:XK 
 
 Hence S'T + TS : S'T - TO = X'N + XN : X'JST - XN, 
 
 2CS : 2GT = 2GN' : 2GX. 
 
 Hence GT, GN^ GS.GX== OJ^^^^^^;^ 
 
128 
 
 THE HYPERBOLA. 
 
 87. Prop. IV. If the tangent at any point P of a 
 hyperbola cut the conjugate axis in t and Pn he the perpen- 
 dicular on the axis ; then will the rectangle Cn . Ct be 
 constant. 
 
 Let Pn produced cut the curve again in P\ and the 
 directrix in M. 
 
 Then, since every chord perpendicular to the conjugate 
 axis is bisected by that axis, it follows, as in Art. 18, 
 Cor. II., that the tangents at P, P' will meet on the 
 conjugate axis, and therefore at the point t. 
 
 Then, we know [Art. 10 and Art. 17] that /S'if and St 
 are respectively the internal and external bisectors of the 
 angle PSP', and are therefore at right angles. 
 
 Hence Z CSt = compt. of z XSM = Z XM8. 
 
 The right-angled triangles CSt and XMS are therefore 
 similar, and we have 
 
 tG:CS = XS:XM==XS:Gn; 
 
 :. tG.Cn = CS,X8 = BG\ 
 
THE HYPERBOLA. 
 
 129 
 
 88. Prop. V. If the normal at a point P of a 
 hyperbola cut the transverse and conjugate axes respectively 
 in the points G, g ; then will the ratio PG : Pg be constant, 
 AlsOyfPN and Pn be the perpendiculars on the transverse 
 and conjugate axes respectively, then will the ratios CG : GN 
 and Gg : Gn be constant. 
 
 Join P to the foci /S, B' of the hyperbola. Draw 
 through P the line PMM perpendicular to and termi- 
 nated by the directrices. Then, since MM and SB' are 
 both bisected by the conjugate axis, it follows that MS, 
 M'S' when produced will meet on the conjugate axis. 
 Let them meet in the point g, and let Pg cut the trans- 
 verse axis in the point G. 
 
 
 9y 
 
 \ 
 
 M \s 
 
 / 
 
 
 
 H 
 
 ^ 
 
 \ 
 
 s'y 
 
 X' 
 
 / 
 
 G / 
 t 
 
 X ( S ] 
 
 \ 
 
 Then m-.MP^Gg'. Pg=^G8' : PM ; 
 
 .-. SG:S'G = PM:PM' 
 
 ^PS'.PS'. 
 
 Hence PGg bisects the angle between SP and S'P 
 produced, and must therefore [Prop. II.] be the normal 
 at P. 
 
 s. c, 9 
 
130 THE HYPERBOLA. 
 
 Then from similar triangles, 
 
 GP:Pg = SM : Mg=:BX : XG; 
 .'. GP:Pg=GS,XS:G8.GX 
 = GB' : GA\ 
 Again GG : GN'= GG : nP=Gg : Pg 
 
 = Sg:Mg = GS:GX. 
 Also Gg:Gn = Gg: GP = Sg : SM 
 
 = SG:SX. 
 
 89. Prop. VI. The circle through the foci of a 
 hyperbola and any point P on the curve passes through the 
 points in which the conjugate axis is cut by the tangent and 
 normal at P. 
 
 Let the circle SPS' cut the conjugate axis in the 
 points t, g ; then these points will clearly be on opposite 
 sides of SS\ Let P and g be on the same side of BS\ 
 
 Then, since tg bisects SS' at right angles, it is the 
 diameter of the circle, and the arcs S% tS must be equal, 
 whence the angles S'Pt, tPS are equal. Hence Pt is the 
 tangent at P, and since tg is the diameter of the circle, 
 tPg is a right angle, and therefore Pg is the normal at P 
 to the hyperbola. 
 
 Cor. Since >Si, S\ g, t are on a circle, 
 gG.Gt = S'G,GS = GS\ 
 Also, since the triangles GGg and TGt are similar, 
 
 GG: Gg=Gt:GT. 
 Thus GG.GT = gG,Gt=S'G,GS = GS'. 
 
THE HYPERBOLA. 131 
 
 90. Prop. VII. If the normal at any point P of a 
 hyperbola cut the transverse and conjugate axes in the 
 points G, g respectively and the diameter parallel to the 
 tangent at P in the point F; then PF.PG = BC^ and 
 PF,Pg = AG\ 
 
 The proof of Art. 75 is applicable. j^^x^^'JEt^ 
 
 of- or^-^ 
 
 EXAMPLES. ^^S;v/^« -^^ --^^^ ^-^ 
 
 1. Given the focus of a hyperbola, the length of its transverse axis 
 and one point on the curve ; find the locus of the other focus and of the 
 centre. 
 
 2. One focus of a hyperbola and the corresponding directrix are 
 given, and it is also known that a given straight line touches the curve. 
 Find the other focus. 
 
 3. Find the locus of the centre of a hyperbola which has a given 
 focus and touches a given straight line at a given point. 
 
 4. One focus of an ellipse is given and also two points on the curve ; 
 find the locus of the other focus and also the locus of the centre. 
 
 5. Shew that the triangles PSg and MPg are similar, and that the 
 ratio Pg : Sg is constant. 
 
 6. Shew that the triangles SPg and GPS' are similar, and that 
 
 PG.Pg = SP,S'P. 
 
 7. Shew that the triangles PST and PtS' are similar, and that 
 
 PT.Pt = SP.S'P. 
 
 8. Shew that the angles PSt and PTS are supplementary. 
 
 9. Find a point P on a hyperbola such that the circle SPS' is least. 
 
 10. Shevftha.tCT.NG=B(P&ndtC.ng=AC^. 
 
 11. If the normal at P cut the conjugate axis in g, the projection of 
 Pg on either focal distance will be equal to half the transverse axis. 
 
 12. Any tangent to a hyperbola cuts the tangents at the extremities of 
 its transverse axis in the points T, T. Shew that the circle whose 
 diameter is TT' passes through the foci. 
 
 9—2 
 
132 
 
 THE HYPERBOLA. 
 
 91. Prop. VIII. If PN he the perpendicular from 
 any point P of a hyperbola on the transverse axis A A', then 
 will the ratio PN^ : AN . A'N he constant. 
 
 Let PA^ PA' cut a directrix in the points K, K' 
 respectively. Join K, K' to the corresponding focus S. 
 
 Then K8 bisects the angle between SP and ^>Si 
 produced, and K'B bisects the angle A'SP [Art. 10]. 
 Hence KS and K'S are at right angles, and therefore 
 
 KX.XK'^XB\ 
 
 Now, from similar triangles, 
 
 PN:AN = KX:AX, 
 and PN'.A'N=^XK' :XA'', 
 
 .-. PN^ : AN .A'N=KX . XK' : AX .XA' 
 = X^:AX.XA'. 
 Thus the ratio PN : AN, A'N, that is 
 PN : CN - CA\ 
 is constant for all positions of P. 
 
THE HYPERBOLA. 133 
 
 Let LS be the semi-latus rectum, then the constant 
 ratio is equal to LS^ : GS'^-GA\ 
 
 Hence PN' : ON' - GA^ = LS' : G8' - CA' 
 
 = LS' : BG' 
 
 = BG':GA\ [Art. 83.] 
 
 Conversely, if iV be any point in the straight line 
 AA' produced, and NP be drawn perpendicular to AA\ 
 and such that the ratio NP' : AN. A'N is constant, the 
 locus of P will be a hyperbola of which A A' is the 
 transverse dxis. 
 
 Ex. 1. N is any point exterior to a given circle and on the fixed 
 diameter AA'. NP is drawn perpendicular to ^^' and equal to the 
 tangent from N to the circle. Shew that the locus of P is a 
 hyperbola. 
 
 Ex. 2. PP' is any chord of a given circle perpendicular to the 
 fixed diameter AA'; shew the locus of the point of intersection of 
 AP and A'P' is a hyperbola. 
 
 92. If the relation 
 
 PN' : GN' - GA' = BG' : GA\ 
 
 be supposed to hold good for all positions of the ordinate 
 PJV, and if Gh be the length of the semi-diameter perpen- 
 dicular to AG A', we shall have 
 
 Gb':-GA'==BG': GA\ 
 
 whence it follows that Gh' = - BG\ or Gb = ^/ -1 . GR 
 
 Thus the imaginary length of the semi-axis conjugate 
 is given by the relation Gb' = -GB'=GA'- GS'; and it 
 should be noticed that the relation between the squares of 
 the lengths of the two axes and the distance between the 
 foci is precisely the same in the hyperbola as in the ellipse. 
 
134 THE HYPERBOLA. 
 
 93. Prop. IX. Shew that the feet of the perpendi- 
 culars from the foci on the tangent at any point of a 
 hyperbola lie on a fixed circle, and that the rectangle 
 contained by these perpendiculars is constant. 
 
 Let SY, S'Y be the perpendiculars from the foci on 
 the tangent at any point P of a hyperbola. 
 
 Join SP, S'P and let SY he produced to meet S'P in 
 H. Join GY. 
 
 Then, since PY bisects the angle SPH, and SYH is 
 perpendicular to PY, it follows that SY= YH and 
 SP^PH. 
 
 Hence S'H = ST-HP = ST-SP = 2GA. 
 
 But, since SG=- GS' and SY= YH, it follows that GY 
 is parallel to S'HP, and GY=iS'H= GA. 
 
 Thus the point F is on the circle whose centre is G 
 and radius GA. 
 
 It can be proved in a similar manner that GY' is 
 parallel to SP and equal to GA, 
 
 Thus the feet of the perpendiculars from the foci on any 
 tangent to a hyperbola are on the circle of luhich the trans- 
 verse axis is a diameter, and which is called the auxiliary 
 circle. 
 
THE HYPERBOLA. 135 
 
 Produce YG to meet the auxiliary circle in Z. Then, 
 since YCZ is a diameter of the auxiliary circle, YY'Z is a 
 right angle. The angle YY'S' is also a right angle, and 
 therefore S'ZY' is a straight line. 
 
 Since GY = GZ, SG = GS' and Z SGY= Z S'GZ, 
 
 it follows that the triangles SGY, S'GZ are equal in all 
 respects, so that 8'Z = SY. 
 
 Hence YS .S'r = S'Z. S'T = 8' A . S'A\ 
 
 Cor. I. If a line through G parallel to the tangent at 
 P cut SP, S'P, produced if ^necessary, in the points E', E 
 respectively ; then PE = PE' =AG. 
 
 For, CE is parallel to PY and GY parallel to S'EP ; hence 
 PE = GY=GA. 
 
 Cor. II. The rectangle contained hy the perpendiculars 
 from a focus on two parallel tangents to a hyperbola is 
 constant. 
 
 The converse of Prop. IX. is important, namely, that 
 if 8 he any point without a given circle whose centre is G 
 and 8 he joined to any point Y on the circle, then the line 
 through Y perpendicular to Y8 will touch the hyperbola of 
 which 8 is a focus and of which the given circle is the 
 auxiliary circle, the point of contact of the tangent being 
 its point of intersection with a line through the other focus 
 parallel to GY. 
 
 EXAMPLES. 
 
 1. The locus of the centre of a conic which has a given focus and 
 touches two given straight lines is a straight line. 
 
 2. One focus of a conic is given and three tangents ; find the other 
 focus. 
 
 3. Shew that the circumcircle of the triangle formed by three 
 tangents to a conic cannot pass through a focus unless the conic is a 
 parabola. 
 
 4. Shew that, if a chord of a given circle subtend a right angle at a 
 given point, the chord will touch a fixed conic of which the given point 
 and the centre of the given circle are the foci. 
 
136 
 
 THE HYPERBOLA. 
 
 94. If Y be the point of contact of either tangent 
 from the focus >Si to the auxiliary circle, the line through 
 Y perpendicular to SY will, by the converse of the last 
 proposition, be a tangent to the hyperbola ; and the line 
 through Y perpendicular to YS will pass through the 
 centre of the conic ; moreover, the point of contact of this 
 tangent will be its point of intersection with a line 
 through the other focus parallel to GY, that is parallel to 
 the tangent itself 
 
 Thus there are two tangents to a hyperbola which pass 
 through the centre of the curve, and the points of contact of 
 these tangents are at an infinite distance from the centre. 
 
 The above result also follows from Art. 86. 
 
 Def. A tangent to any curve whose point of contact 
 is at an infinite distance, is called an asymptote. 
 
 If X be the foot of the directrix corresponding to the 
 focus S, GX : 0F= GY : (7>Sf, whence it follows that the 
 triangles XGY, YG8 are similar, so that GXY is a right 
 angle. 
 
 Hence the asymptotes of a hyperbola pass through the 
 points of intersection of the auodliary circle and either 
 directrix. 
 
THE HYPERBOLA. 
 
 137 
 
 If the tangent at the vertex A cut an asymptote in F, 
 the similar triangles FAG, SYG will be equal in all 
 respects, since CY=GA. 
 
 Thus GF = G8, and therefore 
 
 SY^ = AF^=:GF^ - GA^ = G8'-GA^ = GB". 
 
 The angle between the asymptotes will be greater or 
 less than a right angle according as BG is greater or less 
 than GA. 
 
 When BG = GA the asymptotes are at right angles to 
 one another, and the hyperbola is called a rectangular 
 hyperbola. 
 
 95. It should be noticed that the focal distances of 
 any point on a hyperbola are equal to its distances from 
 the directrices measured parallel to an asymptote. 
 
 For, if P be any point on the curve, and if PM be 
 perpendicular to the directrix XLM and PL be parallel 
 to an asymptote, the triangle LPM will be similar to the 
 triangle YGX. 
 
 Hence LP : PM=GY '.GX=^GA : GX ^SP ; PM) 
 :. LP^BP. 
 
138 THE HYPERBOLA. 
 
 96. Prop. X. The point of intersection of two 
 tangents to a hyperbola which are at right angles to one 
 another is on a fixed circle. 
 
 Let F, Y' be the perpendiculars from the foci S, S^ on 
 any tangent ; then Y, Y' will be on the auxiliary circle. 
 
 Hence, if a tangent perpendicular to YY' cut YY' in 
 T, this tangent will be parallel to SY or S'Y' \ and, since 
 the foci are on opposite sides of any tangent to a hyper- 
 bola, T must be between Y and Y' and therefore within 
 the auxiliary circle. 
 
 Then, if SZ, B'Z' be the focal perpendiculars on the 
 other tangent through T, 
 
 SZ=YZ and S'Z'=TT. 
 
 Hence YT . TY' = SZ . ZS' = BC^ 
 
 But Fy Y' are on the auxiliary circle, 
 therefore BG^ = YT . TY' = GA^ - GT". 
 
 Thus T is on the circle the square of whose radius is 
 GA^ — GB\ This circle is called the Director Circle. 
 
 In the case of a rectangular hyperbola the radius of 
 the director-circle is zero, so that the asymptotes are the 
 only perpendicular tangents. 
 
 The director-circle is imaginary when GB^ is greater 
 than GA^, that is when the angle between the asymptotes 
 is greater than a right angle, and in this case no tangent 
 is at right angles to any other. 
 
THE HYPERBOLA. 139 
 
 97. Prop. XI. The tangents drawn from any point 
 to a hyperbola whose foci are S, S\ are equally inclined 
 to the bisectors of the angle S08\ 
 
 Draw the focal perpendiculars /SF, S'Y' to OQ and 
 SZ, S'Z' to TQ'. Join F, Z and Y\ Z\ 
 
 Then SY, S'Y' are parallel and in opposite directions, 
 and SZ, S'Z' are also parallel and in opposite directions; 
 whence it follows that the angles YSZ and Y'S'Z' are equal. 
 
 But 8Y. S' Y' = BG' = SZ . &Z' ; 
 
 .-. /SfF:>Sf^=>Sf'^':^^'F'. 
 
 Hence the triangles YSZ, Z'B'Y' are similar, and 
 a8ZY=Z.S'YZ'. 
 
 Now S, Y, Z, are cyclic, since SYO and SZO are 
 right angles; and therefore the angles >S^^Fand ;Si OF are 
 either equal or supplementary. 
 
 Similarly & Y'Z' and ^OZ are either equal or supple- 
 mentary. 
 
 Hence ^OF and ^'OZ' are equal or supplementary; and 
 similarly the angles ^OZ, B'OY' are equal or supplementary. 
 
 If the tangents cut the transverse axis in the points 
 Ty T respectively, the points T, T will both be between 
 >Sf and & y and therefore the angles ^OT, 8'OT must be 
 equal. Hence the internal and external bisectors of the 
 angle SOS' are also the internal and external bisectors of 
 the angle TOT\ 
 
140 THE HYPERBOLA. 
 
 Properties of Diameters. 
 
 98. We have already proved that the locus of the 
 middle points of any system of parallel chords of a conic 
 is a straight line through the centre of the conic, which is 
 called a diameter of the conic. We have also proved that 
 the tangents at the extremities of any chord intersect 
 on the diameter which bisects the chord, and that the 
 tangents at the extremities of any diameter are parallel 
 to the system of chords bisected by that diameter. 
 
 There is, however, a very important difference between 
 an ellipse and a hyperbola; for in an ellipse every diameter 
 must cut the curve in two real points, but this is by no 
 means true of every diameter of a hyperbola. 
 
 If the tangent at any point P of a hyperbola cut a 
 directrix in K, we know that PK subtends a right angle 
 at the corresponding focus 8, and therefore SP must be 
 less than PK. 
 
 Now if the diameter parallel to the tangent at P were 
 to cut the curve in Q ; then, since no part of the curve is 
 between the directrices, GQ would cut the directrix in some 
 point L between G and Q. Also, since G is the centre of 
 the curve, QG would cut the curve in another point Q such 
 thate(7 = (;Q'. 
 
 Then, since QG is parallel to the tangent at P, it follows 
 from the definition of the curve that we should have 
 
 SP:PK:=:SQ:QL = SQ' : LQ' = SQ + SQ' : QL + LQ'. 
 
 But SP<PK, and therefore SQ + SQ' < QL + LQ\ 
 which is impossible since L is between Q and Q'. 
 
 Hence the diameter parallel to the ta^igent at any 
 point P of a hyperbola cannot meet the curve in real 
 points. 
 
 Thus of every pair of conjugate diameters of a hyperbola 
 only one will cut the curve in real points. 
 
THE HYPERBOLA. 141 
 
 99. Prop. XII. If the diameter PGP' bisect all 
 chords of a hyperbola parallel to the diameter BCD', then 
 will the diameter BCD' bisect all chords of the conic which 
 are parallel to PCP\ 
 
 See Art. 68. 
 
 100. Prop. XIII. The lines joining any point of a 
 hyperbola to the two extremities of any diameter are 
 parallel to conjugate diameters. 
 
 See Art. 69. 
 
 101. Prop. XIV. If the sides of a parallelogram 
 touch a hyperbola^ its diagonals will be conjugate diameters 
 of the hyperbola. 
 
 See Art. 70. 
 
 102. Measurement of lines. In order to under- 
 stand the full significance of properties of the hyperbola, 
 regard must be paid not merely to the lengths of 
 segments of straight lines but also to the direction in 
 which these segments are measured. 
 
 The direction of any segment is shewn by the order of 
 the letters which define its extremities. Thus AB denotes 
 a length measured from A towards B, and BA denotes the 
 same absolute length measured from B towards A. 
 
 ' When we have to consider different segments of the 
 same straight line, all lengths measured in one direction 
 are considered as positive, and consequently lengths 
 measured in the opposite direction must be considered 
 as negative. It is immaterial, and it is rarely necessary 
 to specify, which of the two directions is taken as the 
 positive direction. 
 
 It is at once obvious that AB + BA — 0. 
 
 The relation AB-\-BG = AG will also hold good for all 
 
 positions oi A, B and C on a straight line. 
 
 The relation in fact only asserts that the step from A to B 
 followed by that from B to C is equivalent to the step from A to C, 
 and this is obviously true wherever B may be with reference to 
 A and G. 
 
142 THE HYPERBOLA. 
 
 Again, the rectangle contained by two segments of 
 a straight line which are in the same direction is 
 considered as positive, and the rectangle contained by two 
 segments of a straight line which are in opposite directions 
 is negative. Thus AB . AG + AB .GA = 0. 
 
 Attention to the above principles will enable us to see 
 more clearly the correspondence between the properties 
 of the ellipse and the hyperbola. 
 
 For example, we have proved in Art. 87 that tC . Cn=BC', tC and 
 Cn heing drawn in the same direction. Hence Gt. Cn = - BG^= Cb^ 
 [Art. 92]. Thus in the hyperbola, as in the ellipse, Ct . Cn is equal 
 to the square of the semi-conjugate axis. 
 
 103. We have already proved in Art. 24 that the 
 ratio of the rectangles contained by the segments of any 
 two intersecting chords of a hyperbola, which are parallel 
 to two given straight lines respectively, is constant for all 
 positions of the point. of intersection; and, by considering 
 lines through the centre of the hyperbola, it follows that 
 the ratio of the rectangles contained hy the segments of any 
 two chords of a hyperbola is equal to the ratio of the squares 
 of the parallel semi-diameters. 
 
 If the rectangle of the segments of one of the chords be 
 positive [see Art. 102] and of the other be negative, one of 
 the semi-diameters parallel to the chords will be real and 
 the other imaginary. But if both rectangles are positive 
 or both negative, the semi-diameters parallel to the chords 
 must be both real or both imaginary. 
 
 104. Prop. XV. If QV he any ordinate to the 
 diameter PGP' of a hyperbola, then will the ratio 
 QV : GV - GP^ be constant. 
 
 Let QF be produced to meet the hyperbola again in 
 Q' i then since QQ' is parallel to the tangent at P, V will 
 be the middle point of QQ\ 
 
 Now, we know [Art. 103] that the ratio of the 
 rectangles contained by the segments of two chords of 
 a hyperbola drawn in given directions is constant. 
 
THE HYPERBOLA. 
 Thus VQ . VQ' : VP . VP' is constant, 
 
 - QY^ : CV^ - CP^ is constant. 
 
 143 
 
 Now, if we suppose the proposition to remain true 
 when the chords intersect in the centre, and if Cd be the 
 semi-diameter parallel to QQ\ which we know is of 
 imaginary length ; then we have 
 
 - QV' : CV - GP' = Cd' : GP\ 
 
 or QV^ : GV^-GP^ = -Gd' : GP\ 
 
 Conversely^ if a line QF be drawn in any fixed 
 direction cutting the fixed line PP' in V so that the ratio 
 QV^ : VP . VP' is constant, then the locus of the point Q 
 will be a h3rperbola or an ellipse according as the point V 
 is without or within the terminated straight line PP'. 
 
 Cor. If on any ordinate Qyto a fixed diameter PP' 
 of a hyperbola a point R be taken such that VR : VQ is 
 constant, the locus of R will be another hj^erbola of which 
 PP' is a diameter. 
 
144 
 
 THE HYPERBOLA. 
 
 105. Prop. XVI. If the tangents at the extremities 
 of a chord QQ' of a hyperbola meet in T, and the diameter 
 CT cut QQ' in V a7id the curve in P ; then will 
 
 CV.GT = GP^. 
 
 If the diameter GT cut the curve in real points, the 
 proof of Art. 71 is applicable without change. 
 
 If the tangents at Q, q, the extremities of a chord 
 which cuts both branches, meet in t, Gt will bisect Qq. 
 And, if QQ' be parallel to Gv, the tangents at Q, Q' will 
 meet at 'T on the diameter PGP' which is parallel to Qq. 
 
 Then vG= QV, and we have 
 
 GT'.TV^Gt'.QV', 
 .'. CV.GT : GV.TV=vG.Gt : QV. 
 Now GV, TV= GV^ - GV. GT = GV' - GP^, by the first 
 .-. vG. Gt : QV'= GP^ : GV^-GP' 
 
 ==-Gd': QV\ [Prop. XY.] 
 
 Hence vG.Ct = - Gd\ or Gv.Gt = Gd\ 
 
 case. 
 
THE HYPERBOLA. 145 
 
 106. Prop. XVII. The length of a focal chord of 
 a hyperbola varies as the square of the parallel semi- 
 diameter. 
 
 The proof given in Art. 79 is applicable whether 
 the focal chord does or does not have its extremities on - 
 the same branch of the curve. 
 
 107. Prop. XVIII. If a circle cut a hypei'hola in 
 four points, the line joining any two of the points and the 
 line joining the other two points will make equal angles 
 with the axes of the hyperbola. 
 
 The proof given in Art. 80 is applicable without 
 change to the hyperbola. 
 
 It should, however, be noticed that since the diameters 
 parallel to the chords are equal they must both be real or 
 both imaginary, from which it follows that both chords or 
 neither wdll cut opposite branches of the hyperbola, and 
 consequently that the four points of intersection of a 
 circle and a hyperbola must all be on one branch or two 
 on each branch of the curve. 
 
 EXAMPLES. 
 
 1. Shew that, if a parallelogram be inscribed in a hyperbola, the 
 diagonals of the parallelogram will be diameters of the hyperbola. 
 
 2. Shew that a circle cannot cnt a hyperbola in more than four 
 points. 
 
 3. If the chords PQ, PQ' of a hyperbola be equally inclined to the 
 transverse axis, the circle PQQ' will touch the hyperbola. 
 
 4. If the normal at any point P of a hyperbola cut the transverse 
 and conjugate axes in G, g respectively, the projections of PG and Pg on 
 either focal distance will be respectively equal to the semi-latus-rectum, 
 and the semi- transverse axis. 
 
 5. Shew that, if QQ' be any one of a system of parallel chords of a 
 hyperbola, the locus of a point P on QQ' such that QP : PQ' is constant, 
 is another hyperbola. 
 
 6. Shew that, if SY, SZ be the perpendiculars from a focus of a 
 hyperbola on two tangents which meet in the point T, YZ will be 
 perpendicular to the line joining T to the other focus of the hyperbola. 
 
 s. a 10 
 
146 
 
 THE HYPERBOLA. 
 
 108. Prop. XIX. If the tangent at any point P of 
 a hyperbola whose foci are S, S' be cut by any pair of 
 parallel tangents in the points T, T' and by an asymptote 
 in the point L ; then will 
 
 FT . PT' = PS . PS' = PL' = PL\ 
 
 Let Q, Q' be the points of contact of the parallel 
 tangents. Join ST, S'T, ST', S'T. On SP take a point 
 H, such that PROPS', and join HT, HT\ 
 
 Then, since TT' bisects the angle UPS' and 
 Pir= PS', it follows that TH = TS' and T'H = TS', and' 
 therefore that the triangles HTT', STT' are equal in all 
 respects. 
 
 Z.HTT'=^ ^S'TT 
 
 = ^STQ [Prop. XL] 
 
 . ^STH= ZT'TQ. 
 
 zST'ff= ^TT'Q\ 
 
 ZT'TQ= zTT'Q\ 
 
 Hence 
 
 Similarly 
 But 
 
THE HYPERBOLA. 147 
 
 since TQ and T'Q' are parallel ; 
 
 Thus >Sf, T', T, H lie on a circle, and therfe^' •* ^S^ - i- ^, 
 
 PT . PT = PE.PS=PS. PS\\^J:f,^^ "' 
 
 Cor. Since the diagonals of a parallelografti whose ^ 
 sides touch a hyperbola are conjugate diameters, the 
 proposition may be enunciated thus: 
 
 If the tangent at P to a hyperbola be cut in T, T' by 
 any pair of conjugate diameters ; then will 
 
 PT . pr = PS . ps\ 
 
 Since an asymptote is in the direction of two coincident 
 conjugate diameters, it follows that if the tangent at P cut 
 an asymptote in L ; then PS . PS' = PL^. 
 
 It is obvious from the above that the portion of any 
 tangent to a hyperbola cut off by any pair of parallel 
 tangents, or by any pair of conjugate diameters, subtends 
 equal angles at the foci. 
 
 EXAMPLES. 
 
 1. If the tangent at any point P of a hyperbola whose foci are S, S' 
 be cut by any pair of conjugate diameters in the points T, T' respectively, 
 then will the rectangle contained by the perpendiculars from T and T' on 
 SP be constant. 
 
 2. If any tangent to a hyperbola be cut by the two perpendi(5ular 
 tangents in the points T, T', the circle through T, T' and either focus 
 will be equal to the auxiliary circle. 
 
 3. If any tangent to a hyperbola be cut by any pair of parallel 
 tangents in the points 2', T\ the circle through 1\ T' and a focus of the 
 hyperbola will never be less than the auxiliary circle. 
 
 10—2 
 
148 THE HYPERBOLA. 
 
 Properties of Asymptotes. 
 
 109. Prop. XX. The portion of amy tangent to a 
 hyperbola intercepted between the asymptotes is bisected at 
 its point of contact. 
 
 Let the tangent at any point P of a hyperbola cut the 
 asymptotes in the points L, L'. 
 
 Join P to the focus 8, and draw SK parallel to the 
 asymptote GL. 
 
 Draw from L and L' the perpendiculars LM, L'M' 
 on 8P. 
 
 Then the asymptote CL is a tangent whose point of 
 contact is at infinity, and therefore LS makes equal 
 angles with PS and SK. [Art. 17.] 
 
 Hence the perpendicular from L on PS is equal to 
 the perpendicular from L on SK, and therefore equal to 
 the perpendicular from S on GY, which is equal to BG. 
 
 [Art. 94] 
 
Thus 
 Similarly 
 
 THE HYPERBOLA. 
 
 LM=Ba 
 L'M' = Ba 
 
 149 
 
 Since the perpendiculars LM, L'M'y are equal, it 
 follows that LP = PL'. 
 
 Ex. 1. The circle circumscribing the triangle formed by any 
 tangent to a hyperbola and the two asymptotes, tcillpass through the 
 points where the corresponding normal cuts the axis. 
 
 Let any tangent cut the asymptotes in L, L', and let tlie circle 
 LCU cut the transverse and conjugate axes in G, g respectively. 
 Let P be the point of intersection of Gg and LL'. 
 
 Then lPLG+ l PGL = L GCL' + z gCL 
 
 = I GCL + L LGg 
 = right angle. 
 
 Hence Gg is perpendicular to LV. 
 
 And, since the diameter Gg of the circle LCL' is perpendicular to 
 the chord LL', it must bisect that chord in P. 
 
 Hence P is the point of contact of the tangent LL', and therefore 
 Gg is the normal at P. 
 
150 THE HYPERBOLA. 
 
 Ex. 2. The portion of any tangent to a hyperbola intercepted 
 between the asymptotes subtends supplementary angles at the foci. 
 
 Let the tangent at P cut the asymptotes in L, L' respectively. 
 Through S draw SK, SK' parallel to the asymptotes, and join SP. 
 Then LS bisects the angle PSK, and L'S bisects the angle PSK\ 
 
 Hence 2 A LSL' = L KSP + Z PSK' 
 
 = 2aCSK=2iS'GL. 
 
 Thus I LSL' = LLCS'. 
 
 Similarly z LS'L' = z LCS. 
 
 Hence the angles LSL', LS'L' are supplementary ; and therefore, 
 as S and S' are on opposite sides of LL', the four points S, S', !•, I.' 
 are on a circle. 
 
 Ex. 3. Shew that the triangles LSL' and LCS' are similar. 
 
 Ex. 4. Shew that the triangles LCS and SCL' are similar, and 
 ihBXLG .GL' = CSK 
 
 Ex. 5. Shew that LS touches the circle GSL'^ and L'S touches 
 the circle GSL. 
 
THE HYPERBOLA. 
 
 151 
 
 110. Prop. XXI. If a double ordinate PNP' to 
 the transverse axis of a hyperbola be produced to cut the 
 asymptotes in R, R\ then will RP . PR' = RP . RP' = BO, 
 
 Let the directrix corresponding to the focus >Si cut an 
 asymptote OR in F; then we know that >SfF is perpen- 
 dicular to the asymptote. 
 
 Draw PK parallel to the asymptote GYR to meet the 
 directrix in K ; then we know that SP = PK, and there- 
 fore ^P=EF. 
 
 Since RR and PP' are both bisected in iV, it is 
 obvious that RP = P'R' and RP' = PR\ 
 
 Hence RP . PR = RP . RP' = RN' 
 
 = SR' 
 
 = SR'- 
 
 -PN^ 
 SP' 
 RY' 
 
 :=^8Y^ = BC\ 
 
 Cor. Since the rectangle RP . PR is invariable, 
 and PR' increases without limit as GB' becomes greater 
 and greater, it follows that RP diminishes without limit 
 as GN is increased. Thus the asymptote approaches 
 indefinitely near to the curve but never crosses it. 
 
152 
 
 THE HYPERBOLA. 
 
 111. Prop. XXII. If a straight line cut a hyper- 
 bola in the points P, F' and the asymptotes in the points 
 T, T' ; then will the rectangle PT . PT' he equal to the 
 square of the parallel semi- diameter of the hyperbola, and 
 PT will he equal to TP', 
 
 Through P draw a straight line perpendicular to the 
 transverse axis and cutting the asymptotes in R^ R' 
 respectively. 
 
 Then we know [Prop. XXI.] that the rectangle 
 PR . PR' is constant and equal to the square of the 
 parallel semi-diameter. 
 
 But if the chord PTT'P' be drawn in any given 
 direction, each of the triangles RPT, R'PT' is of invariable 
 form; and therefore PRiPT is constant, and PR'iPT' 
 is constant for all positions of P. 
 
 Hence the rectangles PT . PT' and PR . PR' are in a 
 constant ratio, and we know that the rectangle PR . PR' 
 is constant. 
 
 The rectangle PT . PT' is therefore constant for 
 all positions of P provided PTT' is drawn in a given 
 directioTi. 
 
THE HYPERBOLA. 153 
 
 If the chord be drawn through the point Q on the 
 hyperbola, such that the diameter QG is parallel to PTT\ 
 it follows that 
 
 PT.PT' = qG\ 
 
 If the chord PTT' cut different branches of the 
 hyperbola, the parallel diameter will cut the curve in real 
 points. 
 
 If, however, a line be drawn through P which cuts the 
 curve in the two points P, p on the same branch of the 
 curve and which cuts the asymptotes in t, t' respectively, 
 the parallel diameter will cut the curve in imaginary 
 points, but the rectangle Pt . Pt' will still be equal to the 
 square of the parallel semi-diameter, the rectangle Pt . Pi! 
 and the square of the parallel semi-diameter being both 
 negative in this case. 
 
 Again, if the straight line PTT' cut the hyperbola 
 again in P' , and Y be the middle point of TT' ; then 
 
 PT.PT' = P'T.P'r; 
 
 :. PV^ — TV^ — P'V^ — T'V\ 
 
 Hence, as TV = VT, PV==yP\ Thus, the middle 
 point of TT' is also the middle point of PP' ; and therefore 
 PT = T'P'. 
 
 As a particular case, when the chord is such that the points P, P' 
 coincide, the middle point of TT' will coincide with either P or P'. 
 This gives another proof of Prop. XX. 
 
 Cor. If the tangent at P cut the asymptotes in L, L' 
 and Gd be the parallel semi-diameter, 
 
 CcZ^ =^PL.PL' = - PL\ 
 since LP = PL'. 
 
154 
 
 THE HYPERBOLA. 
 
 112. Prop. XXIII. The triangle formed hy the 
 asymptotes and any tangent to a hyperbola is of constant 
 
 area. 
 
 Let the tangent at P cut the asymptotes in Z, L'. 
 Draw PiZ, PH' parallel to CL\ GL respectively, and 
 meeting GL, GL' in H, H' respectively. Draw also 
 through P a line perpendicular to the transverse axis 
 cutting GL, GL' in R, R' respectively. 
 
 Since each of the triangles HPR, H'PR' is of in- 
 variable form it follows that PH -.PR and PH' : PR' are 
 both constant, and therefore that PH. PH' : RP . PR' is 
 constant. 
 
 But we know that RP . PR' = BG\ [Prop. XXI.] 
 
 Hence PH . PH' is constant. 
 
 But, since LL' is bisected in P and PH'^ PH are 
 parallel to GL, GL' respectively, GL' = 2HP and 
 GL = 2H'P. 
 
 Hence GL . GL' = 4>PH . PH' = constsmt, 
 and therefore the triangle LGL' is of constant area. 
 
THE HYPERBOLA. 155 
 
 Now, if the tangent at a vertex cut the asymptotes in 
 F, F' respectively, we know that 
 
 CF=GF'=C8. 
 
 Hence GL . CL' = CF . OF' = CS?. 
 
 Also 4Pir . PH' = CL . GL = GS\ 
 
 otherwise thus: — 
 
 Since C is the middle point of SS', 
 
 '2LLGL' = ALS'U - ALUS 
 
 = 2ALPS'-2ALPS 
 
 = ST .BC-SP.BC [Art. 109] 
 
 = 2AC.BC. 
 Or thus : — 
 
 The points L, L\ S, S' are cyclic. [Art. 109, Ex. 2.] 
 
 Let then the circle LSL'S' cut the asymptote CL again in the 
 point I. 
 
 Then, since C is the middle point of SS\ and GL', CI make equal 
 angles with SS' and are on the same side of it, it follows that Cl = CL'. 
 
 Hence GL .GL'=GL . Cl=SG . GS\ 
 
 EXAMPLES. 
 
 1. Construct a hyperbola when the asymptotes and one point on the 
 curve are given. 
 
 2. Construct the hyperbola of which one asymptote and three points 
 are given. 
 
 3. Construct a hyperbola having given one asymptote, two points on 
 the curve and the tangent at one of these points. 
 
 4. Shew that, if a line move in such a manner that the triangle 
 formed by it and two fixed lines is of constant area, the line will always 
 touch a fixed hyperbola. Prove also that the locus of a point which 
 divides in a given ratio the part of the moving line intercepted between 
 the fixed lines, is also a hyperbola of which the fixed lines are asjmiptotes. 
 
 5. Shew that, if two tangents to a hyperbola cut the asymptotes in 
 L, jL' and M, M' respectively ; then will LM' and L'M be parallel to the 
 chord of contact of the tangents and equidistant from it. 
 
 6. Construct a hyperbola having given one asymptote, two tangents 
 and the point of contact of one of them. 
 
156 THE HYPERBOLA. 
 
 113. Prop. XXIV. The sum of the squares of 
 conjugate diameters of a hyperbola is constant 
 
 Let the tangent at any point P of a hyperbola cut the 
 asymptotes in L, L' respectively. 
 
 Then we know that P is the middle point of LL'. 
 
 Hence CZ^ + GL'^ = 2GF' + 2PL\ 
 
 Also, if LK be the perpendicular from L on GL\ 
 
 GD + GL'^ - 2GL' . GK = LL"" = 4^PL\ 
 
 Hence GL . GK = GP^ - PL\ 
 
 But we know that GL . GL' is constant, and the 
 triangle LGK being of invariable form, GK will vary 
 as GL, so that the rectangle GL' . GK is constant. Thus 
 GP^ - PL" is constant. But [Prop. XXII., Cor.] 
 
 PL' = -Gd\ 
 
 where Gd is the semi-diameter parallel to LPL'. Hence 
 GP^ + Gd^ is constant. 
 
THE HYPERBOLA. 157 
 
 114. When a pair of conjugate diameters of a hyper- 
 bola are given, the axes, foci, &c. can be determined. 
 
 For let POP' be the given diameter which cuts the 
 curve in real points, and let KGK' be the direction of the 
 conjugate diameter. Then the diameter KGK' will cut 
 the curve in imaginary points d, d' such that —Gd^ is equal 
 to a given square. 
 
 Then the line through P parallel to KGK' will be the 
 tangent at P ; and, if the points L, L' be taken on this 
 tangent equidistant from P and such that 
 
 LT'^PD=-Gd^ 
 
 then L, L' will be on the asymptotes. Thus the asym- 
 ptotes GL, GU are found, and the axes are the bisectors 
 of the angle LGL\ the transverse axis being along the 
 bisector which is in the same compartment as the point P. 
 Now take points Ky K' on the asymptotes such that 
 
 GK^ = GK'' = GL.GL'; 
 
 then KK ' is the tangent at a vertex of the hyperbola, and 
 GK = G8. Thus the circle centre G and radius GK will 
 cut the transverse axis in the foci. 
 
 115. Prop. XXV. If any line be drawn through a 
 fixed point to cut a conic, the tangents at the points of 
 intersection will meet on a fijxed straight line. 
 
158 THE HYPERBOLA. 
 
 Let QQ' be any chord through the fixed point 0, and 
 let the tangents at Q, Q' meet in T; then TO will bisect 
 QQ' in V. Draw Tt parallel to the diameter conjugate to 
 CO and cutting CO in t. 
 
 First, if CO cut the conic in real points P, P\ Let Pp 
 be drawn parallel to QQ\ and let the tangents at P, p 
 meet at K; then jfiT will be on the diameter CT, and 
 CV.CT=CK.CWyy^±eTe W is the point of intersection 
 ofPpandCZ 
 
 Then, since Tt is parallel to PK, we have 
 Gt: Cp = CT: CK 
 
 = CW : CV, since CV . OT= Cj^ . CW 
 = aP : CO, since PF and OF are parallel. 
 
 Hence CO . Ct = CP^, and therefore t is a. fixed point. 
 
 Next, if CO do not cut the curve in real points, the 
 diameter conjugate to CO will cut the curve in real points, 
 D, D' suppose. 
 
 Let Dd be a chord parallel to QQ' cutting TC in W 
 and OCt in P; then the tangents at D, d will meet in a 
 point K on T(7 such that WC.CK=TC.CV. [Art. 105.] 
 
 Let the tangent at D cut the diameter parallel to QQ' 
 in L. 
 
THE HYPERBOLA. 159 
 
 Then, since GL, GK are conjugate diameters, the 
 rectangle DL . BK is constant for all directions of the 
 chord QQ. [Art. 108.] 
 
 Then FG:GO=WG'.GV 
 
 = GT : KG, since GT. GV= GW. GK, 
 = Gt : KD, 
 since the triangles TGt and GKD have their sides parallel 
 and are therefore similar. 
 
 Hence GO . Gt = FG .KD = -DL.DK= constant. 
 
 Hence t is a, fixed point, and therefore T lies on a fixed 
 straight line. 
 
 Def. The straight line which is the locus of the point 
 of intersection of the tangents at the extremities of any 
 chord of a conic drawn through a fixed point, is called the 
 polar of the point, and the point is called the pole of the 
 line, with respect to the conic. 
 
 The converse theorem can be easily deduced, namely, 
 that if any point be taken on a given straight line and 
 tangents be drawn from it to a conic, the line joining the 
 points of contact of the tangents will pass through a fixed 
 point. 
 
 If the point be without the conic, two lines can be 
 drawn through each of which will cut the conic in 
 coincident points, namely, the two tangents from to the 
 conic; and when the points Q, Q' coincide the point of 
 intersection of the tangents at Q, Q will coincide with 
 them. Thus when a point is without a conic, the tangents 
 at the extremities of any chord through will meet on the 
 line through the points of contact of the tangents from to 
 the conic. 
 
 Cor. I. The polar of a fijxed point with respect to a 
 conic will or will not cut the conic in real points according 
 as the fijxed point is without or within the conic. 
 
 Cor. II. If the polar of a point A with respect to a 
 conic pass through the point B, then will the polar of B 
 pass through A. 
 
160 
 
 THE HYPERBOLA. 
 
 The Conjugate Hyperbola. 
 
 116. Def. Two hyperbolas which have the same 
 asymptotes and whose foci are at the same distance from 
 their common centre are called corrugate hyperbolas. 
 
 Since the axes of a hyperbola bisect the angles between 
 the asymptotes it follows that the axes of two conjugate 
 hyperbolas coincide, the transverse axis of one curve must 
 however lie along the conjugate axis of the other. 
 
 Let P be any point on a hyperbola whose foci are 
 S, S', and let the tangent at P cut the asymptotes in L, L' 
 respectively. 
 
 ^-- 
 
 . 
 
 Q 
 
 ^-.-'^^ 
 
 \ ^N^'^^ 
 
 X/^^^O/p 
 
 S' )/ ^^ 
 
 ^ 
 
 
 x;;^::^^^^*-""^"^ 
 
 . 
 
 -H' 
 
 """~'"'-^-^. \^ 
 
 From L draw the tangent LPV touching the conjugate 
 hyperbola in D, and cutting the asymptote GL' in V. 
 
 Then we know that I> will be the middle point of LV 
 and that GL . CV = (7if ^ where jET is a focus of the 
 conjugate hyperbola. 
 
 Hence LG . CI' =CH^ = GS''^LG .GU \ 
 
 .'. L'G^GV. 
 
 But LD-=DV and LP=^PL'\ therefore GD is parallel 
 to LPL' and GP is parallel to LBV, 
 
THE HYf»ERBOLA. 161 
 
 Thus conjugate hyperbolas have coincident conjugate 
 diameters, and the diameter which cuts one curve in real 
 points lies along the direction of the diameter which cuts the 
 other curve in imaginary points. 
 
 Since PGDL is a parallelogram, PD, and similarly P'D, 
 is parallel to one asymptote and is bisected by the other. 
 
 The area of the parallelogram formed by the tangents 
 at the extremities of the diameter PGP' of one hyperbola 
 and the tangents at the extremities of the conjugate 
 diameter of the conjugate hyperbola is clearly four times 
 the area of the triangle LCL ; and is therefore constant. 
 
 Since Gn = PL, it follows from Art. 113 that 
 CP^ -01)2 is constant. 
 
 Thus the difference of the squares of any diameter of 
 a hyperbola and the conjugate diameter of the conjugate 
 hyperbola is constant. 
 
 The Rectangular Hyperbola. 
 
 117. If a directrix cut the asymptotes GY, GY' of a 
 rectangular hyperbola in the points F, Y' respectively; 
 then, if S be the corresponding focus we know that SYG 
 and 8Y'G are right angles [Art. 94], whence it follows that 
 the figure SYGY' is a square and GS^ = IGY^ = 2GA\ 
 
 Hence the eccentricity of a rectangular hyperbola is sj% 
 
 118. Prop. XXVI. Gonjugate diameters of a rect- 
 angular hyperbola make equal angles with an asymptote. 
 
 Let the tangent at any point P cut the asymptotes in 
 the points Z, L'. 
 
 Since the angle LGL' is a right angle, and P is the 
 middle point of LL' [Art. 109], it follows that P is the 
 centre of the circle LGL' and that the angles PGL and 
 PLG are equal. 
 
 Thus, if Gd be the diameter conjugate to GP, and 
 therefore parallel to LPL', GP and Gd will make equal 
 angles with either asymptote. 
 
 Cor. The angles between any two diameters, or any 
 two chords, of a rectangular hyperbola are equal or supple- 
 mentary to the angles between their conjugate diameters, 
 
 s. c. 11 
 
162 THE HYPERBOLA. 
 
 119. Prop. XXVII. The sum of the squares of two 
 conjugate diameters, or of two perpendicular diameters, of 
 a rectangular hyperbola, is zero. 
 
 Let the tangent at any point P cut the asymptotes in 
 
 Ju, 1j . 
 
 Then we know that the square of Gd, the semi-diameter 
 conjugate to CP, is equal to — PL^. [Prop. XXII.] 
 
 But LGL' is a right angle and P is the middle point 
 of XX'; hence PX = (7P. 
 
 Therefore GP^ + Gd'' = GP^ - PL^ = 0. 
 
 Next, let GP' be the semi-diameter such that the 
 angles P'GA and AGP are equal; and let GP' cut LPL' 
 in F. 
 
 Then ^FaP+ ZFP(7=2Z^CP + 2ZP0X 
 
 = 2 Z J.(7X = a right angle. 
 
 Hence GP' is perpendicular to LPL. 
 Since GP and GP' are equally inclined to the trans- 
 verse axis, GP = GP'. 
 
 Hence GP' ^+Gd'^ GP' + CcZ^ = 0. 
 
THE HYPEKBOLA. 163 
 
 Since there are only two pairs of equal diameters 
 of any conic, which we know are equally inclined to an 
 axis, it follows that, if the sum of the squares of two 
 diameters of a rectangular hyperbola he zero, the diameters 
 must he conjugate or perpendicular. 
 
 Conversely. If a conic he such that (1) the sum of 
 the squares of a pair of conjugate diameters is zero, or 
 (2) if the sum of the squares of two perpendicular diameters 
 is zero ; then in either case the conic must he a rectangular 
 hyperhola. 
 
 The conic must be a hyperhola, since the length of one 
 diameter is real and of the other is imaginary. 
 
 Let P be any extremity of the real diameter, and let 
 the tangent at P cut the asymptotes in L, L'. 
 
 Then in the first case, since 
 
 GF^ = -Gd^ = PL\ GP = LP = PL', 
 
 whence LGL' must be a right angle. 
 
 To prove the second case, if GP be the real diameter, 
 and if KPK' be drawn perpendicular to GP so as to cut 
 the asymptotes of the hyperbola m. K, K' respectively. 
 
 Then we know that PK.PK' is equal to Gd\ the 
 square of the semi- diameter parallel to KPK\ Thus 
 KP . PK' = -Gd'= GP\ by supposition. 
 
 And, since GP is perpendicular to KPK' and 
 
 KP . PK' = GP\ 
 
 it follows that KGK' is a right angle. 
 
 Ex. 1. Construct a rectangular hyperbola having given the centre 
 and any two points on the curve. 
 
 Ex. 2. Construct a rectangular hyperbola having given one 
 asymptote and any two points on the curve. 
 
 Ex. 3. Construct a rectangular hyperbola having given two 
 points on the curve and the tangents at those points. 
 
 11—2 
 
164 ■ THE HYPERBOLA. 
 
 120. Prop. XXVIII. Any chord of a rectangular 
 hyperbola subtends equal or supplementary angles at the 
 extremities of any diameter. 
 
 Let Q, Q' be the extremities of any chord, and PGP' 
 be any diameter of a rectangular hyperbola. 
 
 Join qP, Q'P, QP\ Q'P'. Bisect PQ in V and 
 PQ' in V\ and join CV, CV\ 
 
 Then the angle QPQ' between the chords PQ and PQ' 
 is equal, or supplementary, to the angle VCV between 
 the conjugate diameters CV and GV. 
 
 But, since PQ is bisected in V and PP' in 0, GV 
 must be parallel to P'Q. Similarly GV must be parallel 
 to P'Q\ Hence Z QP'Q' = Z VGV\ 
 
 Therefore the angles QP'Q' and QPQ' are either equal 
 or supplementary. 
 
 Cor. The locus of a point G which moves so that the 
 difference of the angles GBA and GAB is constant, A andB 
 being fixed points, is a rectangular hyperbola of which 
 AB is a diameter. 
 
 For, if <7ie rectangular hyperbola which has AB as a diameter and 
 passes through any one position of the moving point be constructed, 
 it is easy to shew that every other position of the moving point is on 
 this hyperbola. 
 
THE HYPERBOLA. 
 
 165 
 
 121. Prop. XXIX. If a rectangular hyperbola 
 pass through the angular points of a triangle it will also pass 
 through the orthocentre ; and conversely every conic which 
 passes through the angular points and the orthocentre of a 
 triangle must he a rectangular hyperbola. 
 
 Let AD, BE, OF be the perpendiculars of the triangle 
 ABC, and let be the orthocentre. ^^ 
 
 
 Then it is easily seen that the rectangles BD . DC and 
 DO . DA are equal. 
 
 Let a rectangular hyperbola through A, B, C he cut 
 again by DA in the point P. Then, since the sum of the 
 squares of perpendicular diameters is zero, we must have 
 
 DP.DA=-DB.DG = BD.Da 
 
 Thus DO.DA=DP.DA, 
 
 so that F must coincide with 0. 
 
 Conversely, if a conic pass through A, B, and 0, it 
 must be a rectangular hyperbola. 
 
 For since DO .DA =- DB .DC, 
 
 it follows that the sum of the squares of a pair of perpen- 
 dicular diameters is zero, which can only be the case when 
 the conic is a rectangular hyperbola. [Prop. XX VII.] 
 
 Cor. All conies thrvugh the points of intersection of 
 two rectangular hyperbolas are rectangular hyperbolas. 
 
166 THE HYPERBOLA. 
 
 122. Prop. XXX. The locus of the centres of 
 rectangular hyperbolas which circumscribe a triangle is the 
 nine point circle of the triangle. 
 
 Let A, B, be the angular points of the triangle, and 
 let D be its orthocentre. Then we know that every 
 rectangular hyperbola through A, B and G will also pass 
 through D. 
 
 Let U, F, W, K be the middle points of BG, GA, AB 
 and AD respectively. 
 
 A 
 
 Then, since the angle between any two chords is equal 
 or supplementary to the angle between their conjugate 
 diameters, if be the centre of one of the rectangular 
 hyperbolas, the angle VOW is equal or supplementary to 
 the angle GAB which is equal to VJJ W. 
 
 Hence is either on the circle VUW, which is the 
 nine point circle of ABG, or on the circle VA W. 
 
 Similarly must be on the circle KUW, which is the 
 nine point circle, or on the circle KA W. 
 
 The centre must therefore be on the nine point 
 circle of the triangle ABG, for it cannot be at the 
 point A. 
 
 Cor. The angle GAB is equal or supplementary to 
 the angle VO W between the conj ugate diameters accord- 
 ing as A and the centre of the curve are on opposite sides 
 or on the same side of VW. 
 
 Ex. 1. Find the centre of the rectangular hyperbola which 
 passes through four given points. 
 
 Ex. 2. Shew that the nine point circles of the four triangles 
 determined by taking three out of four given points meet in a 
 point. 
 
THE HYPERBOLA. 167 
 
 Limiting Forms of Conics. 
 
 123. We have hitherto considered that the focus of a 
 conic was at a finite distance from the directrix : this may 
 not, however, be the case. 
 
 When the focus is on the directrix, and the eccentricity 
 is greater than unity, it is easy to see that the conic is two 
 straight lines through the focus, and that these straight 
 lines will become more and more nearly coincident as the 
 eccentricity becomes more and more nearly equal to 
 unity. 
 
 Thus two intersecting straight lines may be considered 
 as a hyperbola whose foci and centre are at the intersection 
 of the lines, and whose directrix is either of the bisectors of 
 the angles between them ; also two coincident straight lines 
 may be considered as a parabola. 
 
 It should be noticed that a circle is a conic whose 
 directrix is at infinity, whose foci coincide with the 
 centre of the circle, and whose eccentricity is zero; 
 also that two parallel straight lines may be considered 
 as a parabola whose focus and directrix are both at 
 infinity. 
 
 CoNFOCAL Conics. 
 
 124. Conics whose foci coincide are called confocal 
 conics. 
 
 It is obvious that confocal conics have the same centre 
 and axes. We proceed to consider other important 
 properties of confocal conics. 
 
 When the transverse axis of an ellipse whose foci are given is very 
 great the curve approximates to a circle of infinite radius. 
 
 As the transverse axis becomes more and more nearly equal to the 
 distance between the given foci, the curve becomes flatter and flatter, 
 and the line-ellipse joining the foci is a limiting form of one of the 
 confocals. 
 
168 THE HYPERBOLA. 
 
 When the transverse axis is less than the distance between the 
 foci the conic will be a hyperbola. 
 
 As the transverse axis diminishes indefinitely and ultimately 
 vanishes, the two branches of the hyperbola become more and more 
 nearly coincident with the conjugate axis, and thus the double line 
 coincident with the conjugate axis is a limiting form of one of the 
 confocals. 
 
 Again, as the transverse axis increases and ultimately becomes 
 equal to the distance between the foci, the hyperbola becomes flatter 
 and flatter, and thus a double line coincident with the complement of 
 the line joining the foci is a limiting form of one of the confocals. 
 
 Z. Two conies of a confocal system pass through a given point, 
 of which one is an ellipse and the other a hyperbola. 
 
 For having given the foci S, S' and one point P of an ellipse, the 
 centre G is the middle point of SS', and the major axis is equal 
 to SP + S'P. Hence the vertices ^, ^' are found ; and X, X', the feet 
 of the directrices, are given by the relation GS . GX^GA'^; also the 
 eccentricity is equal to the ratio GS : GA. Thus the ellipse is 
 completely determined, and only one ellipse can be drawn. 
 
 Similarly one and only one hyperbola with S, /S' as foci will pass 
 through the point P. 
 
THE HYPERBOLA. 169 
 
 IZ. One conic of a confocal system and only one will touch a given 
 straight line. 
 
 Let S, S' be the foci and C the middle point of SS'. Draw S Y, the 
 perpendicular on the given straight line. Then OF is equal to the 
 semi-transverse axis of the conic, which is therefore completely 
 determined. 
 
 III. Two confocal conies cut one another at right angles at all their 
 common points. 
 
 For, if two confocal conies intersect at a point P, one must be an 
 ellipse and the other a hyperbola. The tangent to the hyperbola 
 bisects the angle between PS and PS', and the tangent to the ellipse 
 bisects the angle between SP and S'P produced : the tangents to the 
 two confocals are therefore at right angles. 
 
 IV. If a tangent to one of two given confocal conies be perpen- 
 dicular to a tangent to the other, their point of intersection will lie on 
 a fixed circle. 
 
 Let T be the point of intersection of the two tangents TP, TQ ; 
 and let CA, GA' be the semi-transverse axes of the conies on which 
 P, Q respectively lie. 
 
 Draw SY, S'Y' perpendicular to TP; and SZ, S'Z' perpendicular 
 to TQ. Then, the points Z, Z' are on the auxiliary circle of the 
 conic on which Q lies. 
 
 Hence TZ . TZ' = CT^ - GA'\ 
 
 But SY=ZT and S'Y' = Z'T\ 
 
 :. TZ . TZ'=SY . S'Y'=CA^-GS\ 
 
 Hence GT'^ =GA^+GA'^- GSK 
 
 Thus T is on the circumference of a circle concentric with the 
 conies. 
 
170 
 
 THE HYPERBOLA. 
 
 V. The difference of the squares of the perpendiculars drawn 
 from the centre on any parallel tangents to two given confocals is 
 constant. 
 
 Let GK, CK' be the central perpendiculars on two parallel 
 tangents to a conic, and let CL be the perpendicular on a parallel 
 tangent to a confocal. 
 
 Let a line through a focus perpendicular to the tangents meet 
 them in F, Y' and Z respectively. 
 
 Then GU- GK^={GL-^ GK) [GL - GK) 
 
 = K'L. KL=rZ.YZ. 
 
 But Y, Y' are on the auxiliary circle of one conic, and Z is on the 
 auxiliary circle of the other. 
 
 Hence Y'Z . YZ = GZ'^ - GY^ = constant. 
 
 Thus GL^-GK^ is constant, being equal to the difference of the 
 squares of the radii of the auxiliary circles of the two conies. 
 
 VI. The length of a chord of one conic which touches a fixed 
 confocal conic varies as the square of the parallel diameter. 
 
THE HYPERBOLA. l7l 
 
 Let QQ' be any chord of a conic and let the tangents at Q, Q' 
 intersect in T ; and let GT cut QQ' in V and the curve in P. 
 
 Draw GK perpendicular to the tangent at P and let it cut QQ' 
 inL. 
 
 Then, if QQ' touches a fixed confocal conic, we know that 
 GIO - GL^ is constant. 
 
 But, if GD be the semi-diameter parallel to QQ', 
 
 QV^ : GD^=GP^- GV^ : GP^ 
 = GK^ - GL^ : GK^ ; 
 .-. QV^ : GK^-GL^=GD^ : GK^ 
 
 = GD^ : C7D2 . GK^. 
 But GD . GK=AG . EG, 
 
 and GK^ - GL^ is constant if QQ' touches a fixed confocal. 
 Hence QV varies as GD^. 
 
 Conversely, if QF varies as GD^, it follows from the above that 
 GK^- Gl? is constant, and therefore that the chord QQ' touches a 
 fixed confocal. 
 
 Cor. If QQ', a chord of a conic, touches a fixed confocal conic, 
 and Gt he drawn 'parallel to the chord QQ' to meet the tangent at Qint ; 
 then will Gt he of constant length. 
 
 FoY Gt.QV=GD^ 
 
 "VTI. The locus of the poles of a given straight line with respect 
 to one of a system of confocal conies is a straight line perpendicular 
 to the given straight line. 
 
 If the given straight line cut the transverse axis in T, and if 
 a perpendicular line through its pole with respect to any conic whose 
 foci are S, S' and centre G cut the axis in G ; then we know [Art. 82] 
 that GG . GT^GS"". 
 
 Hence (? is a fixed point, and therefore the pole of the given line 
 must lie on a perpendicular line through the fixed point G. 
 
 The given straight line will touch one of the conies of the system, 
 and if P be the point of contact, the pole of the line with respect to 
 the conic it touches will be the point P. 
 
 Thus the locus of the poles of a given straight line with respect to 
 the conies of a confocal system is a perpendicular straight line which 
 is a normal to the confocal which touches the given straight line. 
 
 vm. The envelope of the polar of a given point with respect 
 to a system of confocals is a parahola which touches the axes of the 
 confocals. 
 
]72 
 
 THE HYPERBOLA. 
 
 Let be the fixed point, and let its polar with respect to any 
 conic of the system cut the axes in T, t respectively. Also let the line 
 through perpendicular to its polar cut the axes in G, g respectively 
 and the polar in W. 
 
 Take a line OF such that GT bisects the angle OGF, and take the 
 point F on this line such that GF . GO=GS'^; then i*" is a fixed point 
 for all the conies of the system. 
 
 Since GF . GO = GS\ and GG . GT=GS'^ [Art. 82], 
 
 GG:GO = GF:GT; 
 
 also the angles GGO and FGT are equal. 
 
 Hence the triangles GGO and FGT are similar, and 
 
 Z GFT= L GGO 
 
 = supplement of GtT, 
 
 since the angles tGG and tWG are right angles. 
 
 Hence the points t, G, T, F are cyclic. 
 
 And, since the circle tGT goes through the Jixed point F, it 
 follows that the line tT always touches the parabola whose focus is 
 F and of which GT, Gt are tangents. 
 
 125. We know that the locus of the centres of conies 
 which touch two given straight lines at given points is 
 a straight line. The following is an extension of this 
 theorem : — 
 
THE HYPERBOLA. l73 
 
 The locus of the centres of conies with respect to which the 'poles of 
 two given straight lines are given points^ is a straight line through the 
 intersecticm of the given straight lines. 
 
 I X 
 
 Let OX, or be the given straight lines, and A, 5 the given 
 points. Let G be the centre of one of the conies so that A is the pole 
 of OX, and 5 of OF with respect to the conic centre G. 
 
 Let GA cut OX in a, and GB cut OF in &. 
 
 Since A is the pole of OX, and B is the pole of OF, will be the 
 pole of AB. 
 
 Hence GO and AB are parallel to conjugate diameters of the 
 conic. 
 
 Let AB, BD be drawn parallel to OY, OX respectively. Also let 
 GA, GB cut OY, OX in k, I respectively. 
 
 Then, since BD is parallel to OX, the pole of BD is on GA, and 
 it is also on the polar of B ; hence k is the pole of BD. 
 
 Similarly I is the pole of AD. 
 
 Hence, if Gk, Gl cut BD, AD respectively in K, L, we have 
 
 GK.Gk = CA. Ga and GL.Gl = CB. Cb. 
 Therefore Gk : GA = Ga : GK 
 
 = Cl: GB, 
 
 since BK and al are parallel. 
 Hence kl is parallel to AB. 
 
 But k is the pole of BD and I is the pole of AD, and therefore D 
 is the pole of kl. 
 
 Hence CD and kl are parallel to conjugate diameters, and there- 
 fore GD and AB are parallel to conjugate diameters. 
 
 But we have also proved that CO and AB are parallel to conjugate 
 diameters. 
 
 The lines GO and CD therefore coincide, so that G must lie on 
 the fixed line OD. 
 
174 THE HYPERBOLA. 
 
 EXAMPLES. 
 
 1. From a fixed point S a line SP is drawn to meet a fixed circle in 
 P, and PQ is drawn making an angle SPQ of given magnitude: shew 
 that PQ envelopes a conic one focus of which is at S, and find the 
 position of the other focus. 
 
 2. From any point P on a hyperbola a line is drawn parallel to one 
 asymptote to meet the second asymptote in M, and from a point Q a line 
 is drawn parallel to the second asymptote to meet the first in N. Shew 
 that MN is parallel to PQ. 
 
 3. The tangent at the point P of a hyperbola meets an asymptote in 
 T, TQR is drawn parallel to the other asymptote meeting the curve in Q 
 and the parallel through P to the first asymptote in R. Shew that TR is 
 bisected in Q. 
 
 4. A tangent to a hyperbola at any point P meets an asymptote in 
 T. A line RPR' is drawn parallel to this asymptote and meets a directrix 
 in R' and ST in R, where S is the focus corresponding to the directrix. 
 Shew that P is the middle point of RR'. 
 
 5. A conic of given transverse axis has one focus at the focus of a 
 given parabola and touches the parabola; shew that the conic also 
 touches another parabola coaxial and confocal with the given parabola. 
 
 6. Conies are drawn with a common focus and with transverse axes 
 equal to a given straight line. Shew that the conies all touch two fixed 
 conies. 
 
 7. Find the centre and axes of a rectangular hyperbola having given 
 a focus, one asymptote and another tangent. 
 
 8. Describe the hyperbolas which have a given focus, pass through 
 a given point and have their asymptotes parallel to two given straight 
 lines. 
 
 9. Any straight line is drawn in a given direction to cut two fixed 
 hyperbolas which have the same asymptotes in the points P, P' and Q, 
 Q' respectively. Shew that the rectangle PQ, QP' is constant, 
 
 10. PN is the ordinate of any point P on a rectangular hyperbola, 
 and NQ is a tangent to the auxiliary circle. Shew that PQ passes 
 through a vertex of the hyperbola. 
 
 11. Parallel tangents are drawn to a system of circles which pass 
 through two given points ; shew that the locus of the points of contact of 
 the tangents is a rectangular hyperbola. 
 
 12. Pairs of equal circles are drawn through the points A, B and A^ 
 G respectively. Shew that their other point of intersection is on a 
 rectangular hyperbola through A, B, G and having PC for a diameter. 
 
 13. A point moves so that the lines joining it to two fixed points 
 make equal angles with a fixed straight line. Shew that the locus of the 
 point is a hyperbola. 
 
THE HYPERBOLA. 175 
 
 14. Shew that if an equilateral triangle be inscribed in a rectangular 
 hyperbola, the centre of its circum-circle will be on the curve. 
 
 15. Shew that the locus of the points of intersection of two equal 
 circles which touch two given parallel straight lines at fixed points A, A' 
 respectively, and whose centres are on the same side of AA', is a rect- 
 angular hyperbola. 
 
 16. If any parallelogram be inscribed in a rectangular hyperbola, 
 the rectangle contained by the two perpendiculars drawn from any point 
 of the curve to one pair of parallel sides is equal to the rectangle 
 contained by the perpendiculars drawn from the same point on the other 
 pair of parallel sides. 
 
 17. Points P, Q are taken on a rectangular hyperbola and the conju- 
 gate hyberbola respectively such that PQ subtends a right angle at the 
 common centre. Shew that the locus of the middle point of PQ is 
 another rectangular hyperbola whose asymptotes are the axes of the 
 original curves. 
 
 18. Tangents are drawn to a hyperbola at the points where it is met 
 by any tangent to the conjugate hyperbola; shew that the points of 
 intersection of these tangents will be on the conjugate hyperbola. 
 
 19. Prove that the common chords of a hyperbola and any circle 
 may be grouped in pairs which meet the asymptotes in concyclic points, 
 and that these circles are concentric with the original circle. ' 
 
 20. The normals at the points Q, Q' of a conic intersect at right 
 angles in the point and cut the curve again in the point q, q' respect- 
 ively. Shew that qq' is parallel to QQ'. 
 
 21. A straight line cuts the asymptotes of a conic in R, R' and any 
 pair of conjugate diameters in P, P' ; shew that, if V be the middle point 
 ofP-R', VR^=VP.VP'. 
 
 22. With one focus of a given hyperbola as focus and any tangent 
 to the hyperbola as directrix is described another hyperbola touching the 
 conjugate axis of the former; prove that the two hyperbolas will be 
 similar. 
 
 23. If from any point on a hyperbola a tangent be drawn to its 
 auxiliary circle, the tangent will be equal to the semi-minor axis of the 
 confocal ellipse through the point. 
 
 24. If the tangents at the ends of a chord of a hyperbola meet in T, 
 and TM, TM' be drawn parallel to the asymptotes to meet them in M, M'\ 
 then will MM' be parallel to the chord. 
 
 25. A circle cuts a rectangular hyperbola in the points P, P', Q, Q'. 
 Shew that the tangents at P, P' will meet on the diameter perpendicular 
 to QQ'. 
 
 26. Through a given point P any straight line is drawn meeting two 
 fixed straight lines in the points Q, Q' respectively, and the point P' is 
 taken on the line such that QP=P'Q'. Shew that the locus of P' 
 is a hyperbola. 
 
176 THE HYPERBOLA. 
 
 27. If a line pass through a fixed point, the locus of the middle 
 point of the portion intercepted between two given straight lines is a 
 hyperbola. 
 
 28. TQ, TQ' are tangents to a rectangular hyperbola whose centre is 
 C; shew that, if the bisectors of the angle QTQ' meet QQ' in K, K\ then 
 will CK, GIC be the bisectors of the angle QCQ'. 
 
 29. Through a fixed point a chord PQ of a hyperbola is drawn, 
 and lines PL, QL are drawn parallel to the asymptotes ; shew that the 
 locus of L is a hyperbola whose asymptotes are parallel to those of the 
 given hyperbola and whose centre is at the fixed point 0. 
 
 30. If P be any point on a hyperbola whose foci are S and H, and if 
 the tangent at P meet an asymptote in T, the angle between that asymp- 
 tote and HP will be double the angle STP. 
 
 31. Two tangents to a hyperbola from a point cut a directrix in 
 the points T, T', .and S is the focus corresponding to that directrix. 
 Shew that the circle whose centre is and which touches ST, ST' will 
 cut the directrix in two points the radii to which from the point O are 
 parallel to the asymptotes. 
 
 32. Describe a hyperbola through the angular points of a given 
 parallelogram and having one asymptote in a given direction. 
 
 33. Prove that, if A, B and C be three fixed points, two parabolas 
 can be drawn through A and B with C as focus, and that the axes of 
 these parabolas are parallel to the asymptotes of the hyperbola which can 
 be drawn through C with its foci at A and B. 
 
 34. A circle is described passing through P, P' the extremities of 
 any diameter of a rectangular hyperbola, and cutting the tangent at P in 
 T: prove that P'T and the tangent to the circle at P meet on the 
 hyperbola. 
 
 35. The tangent at a fixed point of a rectangular hyperbola meets 
 GT, GT, any pair of conjugate diameters, in T, T'. Shew that the locus 
 of the centre of the circle GTT' is a straight line. 
 
 36. PP' is any diameter of a rectangular hyperbola and Q is any 
 point on the curve. PR, P'R are drawn at right angles to PQ, P'Q' 
 respectively, intersecting the normal at Q in R, R'. Prove that QR and 
 Q'R' are equal. 
 
 37. The locus of the foci of conies which touch the four sides of a 
 parallelogram is a rectangular hyperbola. 
 
 38. A. circle and a rectangular hyperbola circumscribe a right-angled 
 triangle ABG, C being the right angle, and the tangent to the circle at C 
 meets the hyperbola again in G'; prove that the tangents to the hyperbola 
 at G and G' intersect AB. 
 
 39. Shew that, if the tangents from a poirjt to a given conic make 
 equal angles with a given straight line, the point must lie on a rectangular 
 hyperbola through the foci of the conic. 
 
THE HYPERBOLA. 177 
 
 40. Find the centre of a rectangular hyperbola having given three 
 points on the curve and the tangent at one of them. 
 
 41. OT is the tangent at to a rectangular hyperbola, PQ a chord 
 meeting the tangent at right angles at T; shew that the two bisectors of 
 the angle OCT bisect the lines OP and OQ. 
 
 42. The locus of the points of intersection of tangents to an ellipse 
 which make equal angles with the major and minor axes respectively but 
 are not at right angles, is a rectangular hyperbola whose vertices are the 
 foci of the ellipse. 
 
 43. The locus of the extremities of parallel diameters of a system of 
 co-axial circles is a rectangular hyperbola. 
 
 44. A square is circumscribed to a circle and any tangent to the 
 circle meets two parallel sides of the square in the points P, Q, and a 
 parallel tangent to the circle meets the other two sides of the square in 
 the points R, S. Shew that the points P, Q, R, S lie on a rectangular 
 hyperbola passing through the centre of the circle and whose centre is on 
 the circle. 
 
 45. A chord PP' of a hyperbola cuts the asymptotes in R, R' ; GTV 
 is the diameter bisecting the chord in V, and T is the intersection of the 
 tangents at the extremities of the chord. Prove that the parallelogram 
 described with TV as diagonal and its sides parallel to the asymptotes, 
 has its other corners on the curve and its other diagonal parallel to PP' 
 and a third proportional to PF and PV. 
 
 46. Shew that, if points be taken on a fixed diameter of a central 
 conic, and perpendiculars be drawn from them on their polars, the locus 
 of the feet of these perpendiculars is a rectangular hyperbola. 
 
 47. PP' is a diameter of a rectangular hyperbola, and a circle with 
 centre P and radius PP' cuts the hyperbola again in the points A, B, C. 
 Shew that ABC is an equilateral triangle. 
 
 48. Construct a hyperbola, having given three points on the curve 
 and the directions of the asymptotes. 
 
 49. Through a focus S and the further vertex ^' of a hyperbola 
 whose eccentricity is 2, a circle is drawn cutting the hyperbola in the 
 points A', P, Q, R. Shew that the triangle PQR is equilateral. 
 
 50. If an asymptote and two points of a conic be given, the axes 
 will envelop a parabola. 
 
 51. Shew that, if a conic touch the sides BC, CA, AB of a triangle 
 ABC in P, Q, R respectively, the lines -4P, BQ, CR will meet in a point. 
 
 52. A conic touches the sides BC, GA, AB of a triangle in the points 
 P, Q, R respectively, and the lines QR, PP, PQ cut BC, CA, AB 
 respectively in L, M, N; shew that L, M, N are on a straight line. 
 
 s. c. 12 
 
178 THE HYPERBOLA. 
 
 53. A conic cuts the sides BC, CA, AB of the triangle ABC in the 
 points P, P'; Q, Q' and R, R' respectively. Shew that 
 
 BP . BP' . CQ . CQ' . AR . AR'=BR .BR' . AQ .AQ' .CP . CP'. 
 
 [Carnot's Theorem.] 
 
 54. From any point on one hyperbola tangents are drawn to another 
 having the same asymptotes ; shew that the chord of contact cuts off a 
 constant area from the asymptotes. 
 
 55. A circle intersects a hyperbola in four points; prove that the 
 product of the distances of the four points of intersection from one 
 asymptote is equal to the product of their distances from the other. 
 
 56. Shew that, if a rectangular hyperbola cut a circle in four points 
 the centre of mean position of the four points is midway between the 
 centres of the two curves. 
 
 57. Having given five points on a circle ; shew that the centres of the 
 five rectangular hyperbolas, each of which passes through four of the 
 given points, will all lie on a circle whose radius is half that of the given 
 circle. 
 
 58. From an external point two tangents are drawn to a given conic ; 
 shew that, if the four points where the tangents cut the axes of the conic 
 lie on a circle, the point from which the tangents are drawn must lie on 
 a fixed rectangular hyperbola through the foci of the conic. 
 
 59. If the tangents at the points Q, Q' on a, given conic be at right 
 angles to one another, the line QQ' will always touch a fixed confocal 
 conic. 
 
 60. From any point T, the tangents TP, TP' are drawn to one conic 
 and TQ, TQ' to a confocal conic. Shew that PQ, PQ' are equally inclined 
 to the tangent at P. 
 
 61. PP\ QQ' are the chords of contact of pairs of tangents from a 
 point T to each of two confocal conies, whose foci are S, S\ Prove that, 
 if P, Q, S are collinear, QS'P' and PSQ' will also be coUinear; prove also 
 that the locus of T will be a straight line perpendicular to SS'. 
 
 62. TP is a tangent to one conic, and TQ is a perpendicular tangent 
 to a confocal conic. Shew that the line joining T to the centre of the 
 conies will bisect PQ. 
 
 63. TP is a tangent to a fixed conic and TQ is a perpendicular 
 tangent to a fixed confocal conic. Shew that PQ touches a third fixed 
 confocal. 
 
 64. Tangents parallel to a given straight line are drawn to a system 
 of confocal conies ; shew that their points of contact lie on a rectangular 
 hyperbola through the foci. 
 
 65. A. parallelogram circumscribes a conic and has its sides parallel 
 to two fixed straight lines; shew that the four angular points are on a 
 fixed rectangular hyperbola for all conies of a given confocal system. 
 
THE HYPERBOLA. 179 
 
 66. Shew that, if P be any point on an ellipse and p be the corre- 
 sponding point on its auxiliary circle, an asymptote of the confocal 
 hyperbola through P wiU pass through p. 
 
 67. Tangents are drawn to a given system of confocal conies from 
 any fixed point on the transverse axis. Shew that their points of contact 
 lie on a circle. 
 
 68. If the sides of a triangle which is inscribed in one conic touch a 
 confocal conic, the points of contact will lie on the escribed circles of the 
 triangle. 
 
 69. Shew that, if the line through a point perpendicular to its polar 
 with respect to a given conic pass through a fixed point 0, the polar will 
 envelope a parabola which touches the axes of the given conic. Prove 
 also that the same parabola is obtained if the conic be any one of a given 
 confocal system. 
 
 70. Shew that, if OP, OQhe tangents to a conic, the normals at P, 
 Q and the line PQ will all touch a parabola which touches the axes of 
 the conic. 
 
 71. ABC is a triangle inscribed in an ellipse, and a confocal ellipse 
 touches the sides in A', B', C respectively. Prove that the points A, A' 
 are on the same confocal hyperbola. 
 
 72. A hne through one focus is drawn to meet a system of confocal 
 conies : prove that the tangents to the conies at the points of intersection 
 all touch a fixed parabola. 
 
 73. TQ, TQ' are perpendicular tangents to an ellipse, and TR, TR' 
 are tangents to an interior confocal. Shew that, R, R', O, O' are cyclic, 
 where 0, O' are the intersections of QR', Q'R and Qi?, Q'R' respectively. 
 
 74. Shew that, if tangents TQ, TQ' be drawn from a fixed point T 
 to any one of a system of confocal conies, the circle TQQ' will pass 
 through another fixed point. 
 
 75. Qi Q' are any two points on an ellipse whose foci are S, S'; QS, 
 Q'S' intersect in M and QS', Q'S intersect in N, and the tangents at Q, 
 Q' intersect in T. Shew that M and N are on a confocal hyperbola and 
 that TN, TM are tangents to that hyperbola. 
 
 12—2 
 
CHAPTER V. 
 Sections of a Cone. 
 
 126. Def. The surface generated by a straight line 
 which passes through a fixed point V, and which moves so 
 as always to intersect the circumference of a circle whose 
 plane is perpendicular to the line joining its centre, G, to 
 the point V, is called a right circular cone, of which 
 the point V is called the vertex, and the line CV is called 
 the axis. 
 
 127. Every plane section of a right circular cone is a 
 conic. 
 
 It is easily seen that every section perpendicular to 
 the axis of the cone is a circle. 
 
 Let DAP be any plane section, and let the plane of 
 the paper contain the axis VC of the cone, and be perpen- 
 dicular to the plane BAF, VK and VK' being the 
 generating lines in the plane of the paper. 
 
 Let the cutting plane intersect the plane KVK' in the 
 line ASN. 
 
 Then a sphere can be described (whose centre is that 
 of the circle which touches the three lines VK, VK' and 
 ^iV^ which will touch the plane DAP in some point 8 on 
 AN, and will touch the cone along a circle LRL' suppose, 
 LL' being the diameter of the circle in the plane of the 
 paper. 
 
SECTIONS OF A CONE. 
 
 181 
 
 Let XY be the line of intersection of the cutting plane 
 and the plane of contact of the cone and sphere ; then, 
 since these planes are both perpendicular to the plane of 
 the paper, XYis perpendicular to the plane of the paper 
 and therefore perpendicular to ASN. 
 
 Through any point P on the curve DAP draw a plane 
 perpendicular to the axis of the cone, and let this plane 
 cut ASN in the point N' and the cone in the circle KPK'. 
 Then PN will be perpendicular to ASN and therefore 
 parallel to XF. 
 
 Join PS, and draw PM perpendicular to XY. 
 
 Then, if PF cut the circle LRL' in the point P, PS 
 and PR are both tangents to the sphere and are therefore 
 equal. Also PMXN is a rectangle, and PM=NX. 
 
 Hence SP:PM=PR: NX 
 
 = KL:NX 
 
 =AL:AX 
 
 = AS : AX. 
 
 Hence the curve DAP is a conic whose focus is S and 
 directrix XY. 
 
182 SECTIONS OF A CONE. 
 
 The conic is an ellipse, parabola, or hyperbola according 
 as AS or AL is less, equal, or greater than AX, that is 
 according as the angle LXA is less, equal, or greater than 
 the angle ALX or LL'V. 
 
 Thus the curve is an ellipse or hyperbola according as 
 the cutting plane intersects KA, K'A in two points on 
 the same side or on opposite sides of the vertex F, and the 
 curve is a parabola when the cutting plane is parallel to 
 one of the generating lines of the cone. 
 
 Cor. I. The sections of a given right circular cone hy 
 'parallel planes are conies luhose eccentricities are equal. 
 
 Cor. II. The angle between the asymptotes of a hyper- 
 bolic section of a right circular cone is equal to the angle 
 between the two straight lines in whicli the cone is cut by a 
 parallel plane through its vertex. 
 
 128. Another proof that a plane section of a right 
 circular cone is a conic can be given, provided the section 
 is not parallel to one of the generators of the cone. 
 
 For, let the plane of the paper contain the axis of the 
 cone and be perpendicular to the cutting plane, and let 
 KVK' and kvk' be the generating lines of the cone in the 
 plane of the paper. 
 
 Then, the line of intersection of the cutting plane and 
 the plane KVk is not, by supposition, parallel to either of 
 the generators KK' or kk' ; it will therefore meet them in 
 points A, A' respectively. 
 
 Now two spheres can be described each touching the 
 cutting plane in some point in the line A A' and touching 
 the cone along a circle whose plane is perpendicular to 
 the axis of the cone. [The centres of these spheres will 
 be centres of circles in the plane KVk which touch the 
 three lines A F, A'V and AA'.] 
 
 Let >S^, S' be the points in which these spheres touch 
 the cutting plane, and let LRl, L'R'l' be the circles along 
 which they touch the cone. 
 
SECTIONS OF A CONE. 
 
 183 
 
 Let P be any point on the section. Join PS, PS' and 
 PV, and let PV cut the circles of contact of the spheres 
 in the points R, R' respectively. 
 
 Then PS = PR, since they touch the same sphere. 
 
 Similarly PS' = PR', 
 
 Hence, if A and A' are on the same side of the vertex 
 
 ^ PS + PS' = RR'. 
 
 And, HA, A' are on opposite sides of the vertex (as in 
 the figure) 
 
 PS-PS' = RR'. 
 
 \ 
 
184 SECTIONS OF A CONE. 
 
 But VR and VR' are obviously constant, and therefore 
 RR' is constant. 
 
 Thus the plane section is a conic whose foci are the 
 points of contact of the two spheres which can be inscribed 
 in the cone so as to touch the plane of the section. 
 
 129. From the figure to Art. 127, we see that 
 A'V-AV = A'r-AL = A'S-AS=SS\ 
 
 Also, from the figure to Art. 128, we see that 
 
 A'V+AV=A'l + AL = A'S + AS. 
 
 Hence the vertex of a right circular cone which passes 
 through a given ellipse (or hyperbola) must lie on a hyper- 
 bola (or an ellipse) in a perpendicular plane through its 
 transverse ancis whose foci are the vertices and whose 
 vertices are the foci of the ellipse (or hyperbola). 
 
 130. Def. The surface generated by a straight line 
 which is always perpendicular to the plane of a given 
 circle, and moves so as always to intersect the circum- 
 ference of the circle, is called a right circular cylinder, 
 of which the straight line through the centre of the circle 
 perpendicular to its plane is called the axis. 
 
 A right circular cylinder is therefore the limiting form 
 of a right circular cone as its vertex passes off to infinity. 
 
 It is obvious that all sections of the cylinder by planes 
 perpendicular to its axis are equal circles ; also that any 
 section by a plane parallel to the axis is a pair of parallel 
 straight lines, which become coincident when the plane 
 touches the cylinder. 
 
 It can be proved by the method of Art. 128 that every 
 other section of a right circular cylinder is an ellipse, 
 whose foci are the points of contact of the two spheres 
 which are inscribed in the cylinder so as to touch the 
 plane of the section. 
 
SECTIONS OF A CONE. 185 
 
 EXAMPLES. 
 
 1. Find the locus of the foci of parallel sections of a given right 
 circular cone. 
 
 2. Find the least angle of a cone which it is possible to cut by a 
 plane so that the section is a rectangular hyperbola. 
 
 3. The minor axis of any elliptic section of a cone is a mean propor- 
 tional between the diameters of the circular sections of the cone which 
 pass through the extremities of its major axis. 
 
 4. The minor axes of all elliptic sections of a right circular cylinder 
 are equal. 
 
 131. The method adopted in Art. 127 gives a focus and the 
 corresponding directrix of any plane section of a right circular cone. 
 We may, however, prove that a plane section is a conic without 
 finding a focus or a directrix. Let any cutting plane cut the perpen- 
 dicular plane through the axis of the cone in the line A A', the points 
 A, A' being on the same side of the vertex, F, of the cone. 
 
 Let P be any point on the curve, and draw through P a plane 
 perpendicular to the axis of the cone, and cutting AA' in N. 
 
 Then, this plane will cut the cone in a circle of which KK' is a 
 diameter, K and K' being the points in which the plane cuts the 
 generating lines VA, VA' respectively. Also KK' will pass through 
 N and will be perpendicular to PN, 
 
 Hence PN^=KN.NK'. 
 
 Now each of the triangles KNA, K'NA' has its sides in fixed 
 directions, for all positions of N. Hence, for all positions of P, the 
 ratios KN : AN and NK' : NA', and therefore also 
 
 KN . NK' : AN . NA\ 
 are constant. 
 
186 SECTIONS OF A CONE. 
 
 Hence the ratio PN^ : AN . NA' is constant for all points on the 
 curve, whence it follows that the curve is an ellipse. 
 
 If A, A' be on opposite sides of the vertex, it can be proved in a 
 similar manner that the plane section is a hyperbola ; and also that 
 when the cutting plane is parallel to a generating line of the cone, 
 the section is a parabola. 
 
 132. The following theorem is a generalisation of the 
 focus and directrix definition of a conic :^— 
 
 If a circle touch a conic at P and P\ the extremities of a double 
 ordinate PNP' to the transverse axis, the tangent from any point Q 
 of the conic to this circle is to the perpendicular distance of Q from 
 PP' in the ratio of the eccentricity. 
 
 Let V be the vertex of a right circular cone which passes through 
 the conic AP'A\ AA' being the transverse axis of the conic. Draw 
 a circular section LPL'P' of the cone through the points P, P'; L, L' 
 being on the generating lines VA, VA' respectively. Then a sphere 
 can be described which will touch the cone at all points on the circle 
 LPLP'. The section of this sphere by the plane APA'P' will there- 
 fore be a circle touching the conic at the two points P, P'. 
 
 Let Q be any point on the conic, and let the circular section of 
 the cone through Q cut AA' in M, and VA, VA' in K, K' respectively. 
 
 Let qV cut the circle LPL'P' in R. 
 
 Then the tangent from Q to the circle in which the sphere is cut 
 by the plane APA'P' will be a tangent to the sphere, and will 
 therefore be equal to QR ; also the perpendicular distance of Q from 
 PP' will be equal to MN. 
 
 But QR : MN=KL : MN 
 
 = LA : NA 
 
 = AS : AX, 
 which proves the proposition. 
 
 Cor. The sum or difference of the tangents drawn from any point 
 of a conic to two circles each of which touches the conic at the 
 extremities of a chord perpendicular to its transverse axis is constant. 
 
SECTIONS OF A CONE. 187 
 
 EXAMPLES ON CHAPTEE V. 
 
 1. Shew how to cut a given cone so that the section may be a 
 parabola of given latus rectum. 
 
 2. Shew that, if the vertical angle of a cone be a right angle, the 
 major axis of any elliptic section is equal to the difference of the radii of 
 the focal spheres. 
 
 3. P, P' are the .extremities of any diameter of a given elliptic or 
 hyperbolic section of a right circular cone; shew that the sum of the 
 distances of P and P' from the vertex of the cone is constant. 
 
 4. Shew that, if two sections of a right circular cone have a common 
 directrix, the latera recta of the sections are in the ratio of their eccen- 
 tricities. 
 
 5. Shew that the minor axis of an elliptic section of a right circular 
 cone is a mean proportional to the diameter of the focal spheres. 
 
 6. Shew that the latus rectum of any plane section of a given right 
 circular cone varies as the perpendicular from the vertex of the cone on 
 the plane of section. 
 
 7. If two different sections of a cone have a common directrix, the 
 line joining their foci will pass through the vertex of the cone. 
 
 8. Shew that two elliptic sections of a given cone can be found 
 which have a given point within the cone for focus. 
 
 9. Shew that, if two cones be described so as to touch two given 
 spheres, the ratio of the eccentricity of the two conies in which they are 
 cut by any plane will be constant. 
 
 10. Prove that the axis of every right circular cone which has a 
 given central conic for one of its plane sections will touch a central 
 conic, which is an ellipse or a hyperbola according as the given conic is a 
 hyperbola or an ellipse. 
 
 11. Elliptic sections of a right circular cone are made by planes 
 perpendicular to a given plane containing the axis of the cone, and the 
 ellipses have their minor axes of constant length. Shew that the locus of 
 their centres is a hyperbola. 
 
 12. Two cones which have a common vertex, their axes at right 
 angles, and their vertical angles supplementary, are intersected by a 
 plane at right angles to the plane of their axes. Prove that the distances 
 of either focus of the elliptic section from the foci of the hyperbolic 
 section are equal to the distances of the vertices of the ellipse from the 
 vertex of the cone. 
 
 13. Shew that the locus of the centres of plane sections of a given 
 right circular cone, drawn through a given point on the axis of the cone, 
 is the surface formed by the revolution of a hyperbola about its transverse 
 
188 SECTIONS OF A CONE. 
 
 14. Shew that if the latus rectum of a plane section of a given right 
 circular cone be of given length, the foci lie on the surface generated by 
 the revolution of a hyperbola about its transverse axis. 
 
 15. If 0, O' be the centres of the two spheres inscribed in a right 
 circular cone so as to touch any plane, the sphere on 00' as diameter 
 will pass the auxiliary circle of the section of the cone made by that 
 plane. 
 
 16. Shew that the director-circle of any plane section of a cone lies 
 on the sphere which passes through the two circles of contact of the focal 
 spheres. 
 
 17. Prove that a right cylinder on a given elliptic base can be cut in 
 two ways so that the curve of section may be a circle ; and that a sphere 
 can always be drawn through any two circular sections which are not 
 parallel. 
 
 18. A surface is formed by the revolution of an ellipse about its 
 major axis, and a plane is drawn cutting the surface and touching at a 
 point S a sphere inscribed in the surface. Shew that the section is an 
 ellipse of which /S is a focus. 
 
 19. Shew that the centres of elliptic sections of a right circular 
 cone which have major axes of equal length lie on the surfaces generated 
 by the revolution of an ellipse about one of its axes. 
 
 20. Shew that the locus of the vertices of the right circular cones 
 which pass through one conic is a conic of the other species in a perpen- 
 dicular plane, whose vertices are the foci and whose foci are the vertices 
 of the former ; and deduce that the sum or difference of the distances of 
 a variable point on one of these conies from any two fixed points on the 
 other is constant. 
 
CHAPTER VI. 
 
 Orthogonal Projection. Similarity of Curves. 
 Curvature of Conics. 
 
 133. Def. The foot of the perpendicular from a point 
 on a fixed plane is called the orthogonal projection of 
 the point on that plane, and the fixed plane is called the 
 plane of projection. 
 
 If a point describe any curve its orthogonal projection 
 on a given plane will describe a curve which is called the 
 orthogonal projection of the given curve. 
 
 In general, if any point P be joined to a fixed point V, and VP be 
 cut by any fixed plane in the point P', the point P' is called the 
 projection of P on the fixed plane ; also the point V is called the 
 centre of projection, and the fixed plane is called the plane of pro- 
 jection. 
 
 Thus orthogonal projection is only a particular case, when the 
 centre of projection is at an infinite distance and in a direction 
 perpendicular to the plane of projection. 
 
 134. The principal properties of orthogonal projection 
 are the following : — 
 
 (i) The projection of a straight line is a straight 
 line. 
 
 For, let the given line cut the plane of projection in the point A^ 
 and let P' be the projection x)f any point P on the given line. Then, 
 if Q be any other point on the given line, and if QQ' be the perpen- 
 dicular from Q on the line AP', then QQ' will be parallel to PP' and 
 therefore also perpendicular to the plane of projection, so that Q' is 
 the projection of Q. Thus the projection of every point on AP will 
 lie on the line AP'. 
 
190 ORTHOGONAL PROJECTION. 
 
 (ii) Parallel straight lines project into parallel 
 straight lines. 
 
 For the projection of the point of intersection of two straight 
 lines is the point of intersection of their projections, and if one of 
 these points is at infinity the other must also be at infinity. Hence, 
 if the original lines are parallel, the projected lines will be parallel ; 
 and, conversely, if the projections are parallel the original lines must 
 have been parallel. 
 
 (iii) Parts of the same straight line, or of parallel 
 straight lines, are in the same ratio as their projections. 
 
 For let A'B' and CD' be respectively the projections of the 
 parallel straight lines AB and CD. 
 
 Draw lines through 'A and G parallel to A'B' or CD' and meeting 
 BB', CC respectively in the points X, Y. 
 m 'Then the triangles XAB, YCD will be similar, and therefore 
 
 AB :AX=CD : GY; 
 
 .-. AB : GD = AX : GY 
 
 =A'B' : G'D', 
 
 Bince AX= A'B' and GY=G'D'. 
 
 (iv) The number of points in which a curve is cut by 
 a straight line (or one plane curve is cut by another) is 
 equal to the number of points of intersection of their 
 projections. 
 
 (v) The projection ■ of a tangent to a curve is a 
 tangent to the projection of the curve. 
 
 For, if two points of intersection of a straight line and a curve be 
 coincident, two points of intersection of their projections will also be 
 coincident. 
 
 Conversely, if the projections of a straight line and a curve touch 
 one another, the straight line and the curve must themselves touch. 
 
 (vi) The area of any curve on a given plane and of 
 
 its projection on another given plane are in a constant 
 
 ratio. 
 
 Divide the given area into any number of rectangles by two sets 
 of equidistant lines parallel and perpendicular respectively to the 
 line of intersection of the given plane and the plane of projection. 
 Then, those segments which are parallel to the line of intersection 
 • will be unaltered by projection, and those which are perpendicular 
 will be diminished in a constant ratio. [This ratio will be 1 : cos ^, 
 
ORTHOGONAL PROJECTION. 
 
 191. 
 
 where 9 is the angle between the planes.] Hence every rectangle, and 
 therefore the sum of any number of rectangles, will be diminished 
 by projection in a constant ratio. But when the parallel lines are 
 drawn indefinitely near together, so that each of the rectangles is 
 made indefinitely small, their sum is ultimately equal to that of the 
 area in which they are drawn. Hence any area in a given plane is 
 in a constant ratio to the area of its projection on any other given 
 plane. 
 
 135. The projection of a circle is an ellipse. 
 
 Let LM be the line of intersection of the plane of the 
 circle and the plane of projection. 
 
 Let AG A' be the diameter of the circle which is 
 parallel to LM, and let BOB' be the perpendicular 
 diameter. 
 
 Let aca' and bcb' be the projectit)ns of AG A' and BGB' 
 respectively; then since AG A' is parallel to the plane of 
 projection, aca' =AGA\ and bcb' is perpendicular to aca\ 
 
 Let NP be any ordinate to the diameter AG A' of the 
 circle, and let np be its projection. Also, let np cut the 
 circle on aa' as diameter in the point q. 
 
 Then, since the circles ABA' and aqa are equal and 
 en = GN, nq must be equal to NP. 
 
192 
 
 ORTHOGONAL PROJECTION. 
 
 Now np and cb are the projections of the parallel 
 straight lines NP and GB respectively ; 
 
 .-. np :cb = NP :BG 
 
 = nq : ca. 
 
 Hence np : nq = cb : ca, 
 
 whence it follows that the locus of p is an ellipse of which 
 the circle aqa' is the auxiliary circle, 
 
 136. An ellipse can he projected into a circle. 
 
 Let AGA\ BGB' be the major and minor axes of an 
 ellipse. 
 
 Draw a plane through A A' perpendicular to BGB', and 
 in this plane describe a circle on AA' as diameter, and let 
 the chord AK oi the circle be equal to BGB'. 
 
 In the plane of the ellipse draw any line LM parallel 
 to BGB' and cutting the major axis in the point X. Then, 
 if XFbe parallel to AK, the projection of the ellipse on 
 the plane LMY will be a circle equal to the minor 
 auxiliary circle of the given ellipse. 
 
 For, since XF is parallel to AK, XY must lie in the 
 
ORTHOGONAL PROJECTION. 193 
 
 plane A'AK. And, since BOB' is perpendicular to the 
 plane A'XY, so also is the parallel line LXM. Hence 
 a, a' the projections of A, A' respectively on the plane 
 LMY, will lie on the line XY. 
 
 Let h, h' be the projections of B, B' respectively ; then, 
 since BOB' is parallel to the plane of projection, 
 
 hch' = BGE', 
 
 also hch' is perpendicular to aca. 
 
 Let NP be any ordinate to the diameter BOB' of the 
 ellipse, and let np be the projection of NP. 
 
 Then, since NP is parallel to GA, np is parallel to ca, 
 and 
 
 np:ca = NP:GA 
 
 = NQ:BG, 
 
 if Q be the point where NP cuts the minor auxiliary 
 circle of the ellipse. 
 
 Hence, as ca is by construction equal to BG, np = NQ. 
 But en = GN, and np is perpendicular to en, whence it 
 follows that the locus oi p is a. circle equal to the minor 
 auxiliary circle of the ellipse. 
 
 •i 
 
 137. If a central conic be orthogonally projected into 
 any other conic; then, since every chord through the centre 
 of the original conic is bisected in that point, and the 
 segments of a straight line are in the same ratio as their 
 projections, it follows that the projeetion of the eentre of 
 the original conic will he the centre of the projection. 
 
 Also, since tangents project into tangents and parallel 
 lines into parallel lines, it follows that any pair of conju- 
 gate diameters of tlie original conic will project into a pair 
 of conjugate diameters of the projection. 
 
 Ex. 1. Shew that every orthogonal projection of a conic is a 
 conic of the same species whose centre is the projection of the centre 
 of the original conic. [Use Art. 45, Art. 81 or Art. 104.] 
 
 Ex. 2. Shew that any two intersecting straight lines can be 
 orthogonally projected into perpendicular straight lines. 
 
 s. c. 13 
 
194 ORTHOGONAL PROJECTION. 
 
 Ex. 3. Shew that any hyperbola can be orthogonally projected 
 into a rectangular hyperbola. 
 
 Ex. 4. Shew that the ratio of the area of an ellipse to the area 
 of its auxiliary circle is equal to the ratio of the minor to the major 
 axis. 
 
 138. Many of the properties of an ellipse may be 
 proved by projecting the ellipse into a circle. These are 
 called projective properties. 
 
 Ex. 1. The locus of the middle points of parallel chords of an 
 ellipse is a straight line. 
 
 For project the ellipse into a circle. 
 
 Then the system of parallel chords will project into a system of 
 parallel chords, and the middle points of the original chords will 
 project into the middle points of the projected chords. 
 
 Thus we have to prove that the locus of the middle points of 
 parallel chords of a circle is a straight line, which follows at once 
 from Euclid iii. 3. 
 
 Ex. 2. If the tangents at the extremities of the chord QQ' of an 
 ellipse whose centre is G meet in the point T, and GT cut QQ' in V 
 and the ellipse in P, then will CV . GT= GP^. 
 
 Project the ellipse into a circle; then the centre of the ellipse 
 will project into the centre of the circle, for every chord of the circle 
 through the projection of the centre of the ellipse will be bisected at 
 that point. 
 
 Let c, t, q, q', v be the projections of G, T, Q, Q', V respectively ; 
 then since a tangent to any curve will project into a tangent to 
 the projection of the curve, tq and tq' will touch the circle. 
 
 Also, since parts of the same straight line are in the same ratio 
 as their projections; 
 
 GV : GP = cv:cp, 
 
 and GP : GT=cp : ct. 
 
 But, since the triangles cvq and cqt are similar, 
 
 CV : cq = cq : ct; 
 
 CV : cp — cp : ct. 
 
 Hence GV : GP=GP : GT, 
 
SIMILARITY OF CURVES. 195 
 
 Similarity of Curves. 
 
 139. Def. Two curves are said to be similar and 
 similarly situated when radii drawn to the first from a 
 certain fixed point are in a constant ratio to parallel 
 radii drawn to the second from another fixed point 0\ 
 
 Two curves are similar^ but not similarly situated, 
 when radii making a constant angle with one another, 
 drawn to the curves from two fixed points 0, 0' respec- 
 tively, are proportional. 
 
 The two fixed points 0, 0' are called centres of 
 similarity; and when 0' coincides with the point is 
 called a centre of similitude of the two curves. 
 
 140. If one pair of centres of similarity exist for two 
 curves, there will he an infinite number of such pairs, and 
 if the curves he similarly situated a centre of similitude 
 ca7i he found. 
 
 Let 0, 0' be the given centres of similarity, and let 
 OP, O'P' be any pair of corresponding radii. Take G any 
 point whatever, and draw O'C making the same angle 
 with O'P' that OC makes with OP, and let G' be such that 
 the ratio O'G' : OG is equal to the constant ratio O'P' : OP. 
 Then the triangles GOP, G'O'P' will be similar and there- 
 fore G'P' will make the same angle with GP that O'P' 
 makes with OP, moreover G'P' : GP will be equal to the 
 constant ratio O'P' : OP. Hence G and G' are centres of 
 similarity for the two curves. 
 
 Again, if the curves be similarly situated and OP, O'P' 
 be any pair of parallel radii ; then, if PP' cut 00' in the 
 point A, OA : 0'A = OP : O'P' = constant. Hence -4 is a 
 fixed point for all directions of the parallel radii; also 
 AP : AP' = OP : O'P' = constant. Hence ^ is a centre 
 of similitude. 
 
 141. If two curves he similar the tangents at corre- 
 sponding points will he inclined at a constant angle to one 
 another. 
 
 13—2 
 
196 SIMILARITY OF CURVES. 
 
 Let 0, 0' be centres of similarity of the two curves, and 
 let OP, O'P' and OQ, O'Q' be any two pairs of corresponding 
 radii. 
 
 Then the angles POQ, FO'Q' will be equal and 
 OP:0'F=^OQ:0'Q'. 
 
 Hence the triangles POQ, P'O'Q are equiangular, and 
 therefore the angle between PQ and P'Q' is equal to the 
 constant angle between OP and 0'P\ When OQ and 
 OQ' move up to coincidence with OP and OP' respectively, 
 PQ and P'Q' ultimately become the tangents at P and P' 
 respectively. 
 
 142. If two central conies he similar and similarly 
 situated the centres of the two curves will he centj^es of 
 similarity, and the conies must he of equal eccentricity. 
 
 Let and 0' be two centres of similarity of the two 
 curves. Draw any chord POQ of the one, and the parallel 
 chord P'O'Q' of the other. Then by supposition 
 
 PO.OQ: P'O' . O'Q' 
 
 is constant for every pair of corresponding chords. But, 
 since is a fixed point, PO . OQ is always in a constant 
 ratio to the square of the parallel semi-diameter of the 
 first conic; and the same applies to the other conic. 
 Hence parallel diameters of the two conies are in a con- 
 stant ratio to one another. 
 
 Since parallel diameters of the two conies are in a 
 constant ratio to one another, the greatest or least 
 diameter of one conic must be parallel to the greatest or 
 least diameter of the other, and therefore the axes of the 
 two conies must be parallel. 
 
 Also, if GP, G'P' be parallel diameters of the two 
 curves, the tangents, and therefore the normals, at P, P' 
 must be parallel ; whence it follows that 
 
 CO:GN=G'0' :C'N', 
 
 which shews that the eccentricities of the two conies are 
 
SIMILARITY OF CURVES. 197 
 
 The converse theorem, namely that conies of equal 
 eccentricity are similar curves can be easily proved, the 
 foci being centres of similarity. • 
 
 143. If two curves he similar and one of them is a 
 conic the other must also he a conic. 
 
 Let 0, 0' be the centres of similarity of the two curves, 
 and let OP, O'P' be corresponding radii. Let S, H be 
 the foci of the conic, and join BO, HO. Draw the lines 
 0'B\ O'H' such that the angles SV'P\ H'O'P' may be 
 equal to the angles SOP, HOP respectively. Then, since 
 OS and OH are fixed in direction, O'S' and O'H' are also 
 fixed in direction. Hence, if the points S', H' be such 
 that O'S' : OS = O'H' : OH = O'F : OP = constant, the 
 points S' , H' will be fixed. 
 
 The triangles SOP, S'O'P' will be similar, and the 
 triangles HOP, H'O'P' will be similar. 
 
 Hence SP:S'P' = OP: O'P', 
 
 and HP:H'P'=OP: O'P'; 
 
 .-. SP±HP: S'P' ± H'P' is constant, 
 
 whence it follows that, as P describes a conic whose foci 
 are S and H, P' will describe a conic whose foci are S' 
 and H'. 
 
 The above holds good for all positions of the foci of 
 the given conic, and therefore it will hold good in the 
 limiting case when one of the foci is at an infinite dis- 
 tance. Thus the case when the conic is a parabola needs 
 no separate investigation. A modified proof may, however, 
 be given. 
 
 Ex. 1. The middle points of all chords of a conic which pass 
 through a fixed point is a similar and similarly situated conic. 
 
 Let G be the centre of the conic, and let GO cut the conic in the 
 point A. 
 
 Let PP' be any chord of the conic through the fixed point and 
 let V be the middle point of PP'. 
 
 Draw the chord A A' of the conic parallel to PP' and let W be the 
 middle point of AA'. 
 
198 CURVATURE OF CONICS. 
 
 Then we know that CVW is a straight line. 
 
 Since AW=^AA', the locus of W is a, curve similar and similarly 
 situated to the locus of A' ; the locus of W is therefore a conic similar 
 and similarly situated to the given conic. 
 
 Again, since A W and V are parallel, 
 
 CV : GW=GO : C^ = constant. 
 
 Hence the locus of V is similar and similarly situated to the locus 
 of W ; the locus of V is therefore a conic similar and similarly 
 situated to the original conic. 
 
 Ex. 2, The locus of the point of intersection of the tangents at the 
 extremities of any tivo conjugate diameters of an ellipse is a similar 
 and similarly situated ellipse, 
 
 Ex. 3. A chord of an ellipse passes through the ends of two 
 conjugate diameters; shew that its middle point is on a similar and 
 similarly situated ellipse. 
 
 Curvature. 
 
 144. The curvature of a curve at any point is the 
 rate of its bending at that point. Now the curvature of a 
 circle is obviously the same at all points, moreover the 
 rate of bending of a circle changes continuously from 
 infinity to zero as the radius of the circle changes from 
 zero to infinity. Thus there is one and only one circle 
 whose curvature is the same throughout as the curvature 
 of a given circle at a given point. 
 
 If a circle be drawn through any three adjacent points 
 P, Q, i? of a curve, and if the points P, R move up to and 
 ultimately coincide with the point Q ; then the curvature 
 of the circle in its ultimate position will be equal to that 
 of the curve at the point Q ; also this circle in its 
 ultimate position may be said to touch the curve at Q 
 and cut it in another point indefinitely near to the 
 point Q. 
 
 We have therefore the following definitions : — 
 
 Def. If a circle touch a curve at a point P and cut 
 the curve at an adjacent point Q, and if the radius of the 
 circle be made to change in such a way that the point Q 
 
CURVATURE OF CONICS. 199 
 
 moves up to and ultimately coincides with P, then the 
 circle in its ultimate position is called the circle of 
 curvature of the curve at the point P. 
 
 Also the centre of the circle is called the centre of 
 curvature of the curve at P, and the chord of the circle 
 in any direction is called the chord of curvature of the 
 curve in that direction. 
 
 145. To find the chord of curvature at any point of a 
 central conic in any direction. 
 
 Let a circle be drawn touching the conic at the given 
 point P and cutting it at an adjacent point Q. 
 
 Let the diameter PGP' cut the circle in p, and let a 
 line through Q parallel to PGP' cut the conic again in Q', 
 the circle in g, and the tangent at P in T. 
 
 Then when Q moves up to and ultimately coincides 
 with P, Pp will ultimately be the chord of -curvature at P 
 through the centre of the conic. 
 
 Now from the conic we have 
 
 GP':GD'^ = TQ.TQ':TP% 
 
 where GD is the semi-diameter conjugate to GP. 
 But, since TP touches the circle, TP^^'TQ. Tq. 
 
 Hence GP' : GD' = TQ . TQ' iTQ.Tq 
 
 = TQ':Tq. 
 
200 CURVATURE OF CONICS. 
 
 But when the circle becomes the circle of curvature at 
 P, the chords QQ'q and PP'p will coincide, so that 
 TQ' = PP' and Tq = Pp ultimately. 
 
 Hence we have ultimately 
 
 GP':CD'=^PP':Pp=::2GP:Pp; 
 
 .'.Pp.GP = 2GI)' (i). 
 
 Thus the chord of curvature through the centre of the 
 conic is equal to 2GD^IGP. 
 
 If PK be the diameter of the circle, it will cut GD at 
 right angles in the point F, and KpP will also be a right 
 angle. 
 
 Hence PK.PF = PG.Pp 
 
 = 2Ci)^ from (i). 
 
 Thus the diameter of curvature is equal to 2GD^jPF. 
 
 Again, if PI be the chord of curvature in any other 
 direction, and if PI cut GD in the point L ; then, since 
 PIK is a right angle 
 
 PLPL^PF,PK = 2GD'. 
 
 If PI pass through a focus of the conic, we know that 
 PL = GAy so that the chord of curvature through a focus 
 of the conic is equal to ^GD'^jGA. 
 
 146. To find the chord of curvature at any point of 
 a parabola in any direction. 
 
 Let a circle be drawn touching the parabola at the 
 given point P and cutting it at an adjacent point Q. 
 
 Let the diameter through P cut the circle in p, and 
 let the diameter through Q cut the circle in q and the 
 tangent at P in T. 
 
 Let QFbe the ordinate to the diameter PVp oi the 
 parabola. 
 
 Then, since TP touches the circle, 
 
 TP'^ = TQ.Tq. 
 
CURVATURE OF CONICS. 
 
 201 
 
 But PVQT is a parallelogram, and therefore TF = QV 
 andTQ^PF. 
 
 Hence QV' = PV.Tq, 
 
 whence it follows that Tq = 4>SfP, where 8 is the focus of 
 the parabola. 
 
 But when the circle becomes the circle of curvature at 
 P, the diameters Pp and Qq will coincide, so that 
 Tq — Pp ultimately. 
 
 Hence the chord of curvature through P parallel to the 
 axis of the parabola is equal to 4<SP. 
 
 Since PS and PV make equal angles with the 
 tangent to the circle, the chord of curvature through the 
 focus must also be equal to 4>SiP. 
 
 If PK be the diameter of curvature, and ^SfF be the 
 focal perpendicular on the tangent at P, the triangles 
 KPp and PSY will be similar, so that 
 
 P^:Pp = ^P:;SfF. 
 
 Thus the diameter of curvature is equal to ^SP^jSY. 
 
 147. We know that, if a circle cut a conic in four 
 points the line joining any two of the points and the line 
 joining the other two points make equal angles with an 
 axis. 
 
 Hence, if a circle touch a conic at a point P and cut it 
 in two other points Q, R ; then the tangent at P and the 
 
202 CURVATURE OF CONICS. 
 
 chord QR make equal angles with an axis. Also, if the 
 radius of the circle be changed so that the point Q moves 
 up to and ultimately coincides with P, then the tangent 
 at P and the chord PR will make equal angles with 
 an axis. 
 
 Thus the tangent at any point P of a conic and the 
 common chord of the conic and its circle of curvature at P 
 make equal angles with an axis. 
 
 The common chord of a conic and its circle of 
 curvature at any given point can therefore be at once 
 drawn. 
 
 EXAMPLES. 
 
 1. Find a point P on a parabola such that the circle of curvature at 
 F will pass through the other extremity of the focal chord through P. 
 
 2. Prove that the distance of the centre of curvature at any point of 
 a parabola from the directrix is three times that of the point. 
 
 3. Shew that the common chord of a parabola and the circle of 
 curvature at any point on the curve is a tangent to another fixed 
 parabola. 
 
 4. Find the points on a parabola the circles of curvature at which 
 pass through a fixed point on the curve. 
 
 5. Having given a point on a parabola and the circle of curvature 
 of the curve at that point, shew that the focus must lie on a fixed circle 
 and that the directrix must pass through a fixed point. 
 
 6. Find a point P on an ellipse such that the circle of curvature at 
 P will pass through the other extremity of the diameter through P. 
 
 7. Shew that the diameter of curvature of a conic at an extremity of 
 its transverse axis is equal to the latus-rectum. 
 
 8. P is any point on a parabola whose vertex is A, and PM, FN are 
 the perpendiculars on the directrix and axis respectively. Shew that, if 
 MA and FN intersect in the point 0, the diameter through will pass 
 through the centre of curvature at P. 
 
 9. Shew that, if S be the focus of a parabola, P any point on the 
 curve and the centre of curvature at that point; then 
 
 FO^=SO^ + SPSr-. 
 
 10. Shew that, at any point P of a conic the chord of curvature 
 through a focus is equal to the focal chord of the conic which is parallel 
 to the tangent at P. 
 
EXAMPLES. 203 
 
 11. The normal at P to a rectangular hyperbola cuts the curve 
 again in Q ; shew that PQ is equal to the diameter of curvature at P. 
 
 12. Prove that the circle of curvature of an ellipse at P, one of the 
 extremities of an equi-conjugate diameter, cuts the auxiliary circle so 
 that their common chord is parallel to the diameter through P. 
 
 13. Construct an ellipse having given one focus, one point on the 
 curve and the circle of curvature at that point. 
 
 14. Shew that the centre of curvature at any point P of an ellipse is 
 the pole of the tangent at P with respect to the confocal hj'perbola 
 through P. 
 
 15. Shew that, if the circle of curvature at any point P on an ellipse 
 cut the curve again in 0, there will be two other points Q, R on the 
 ellipse the circles of curvature at which will pass through 0, and the four 
 points P, Q, R, will lie on a circle. 
 
 16. The circle of curvature of a parabola at P meets the curve 
 again in Q, and the tangent at P cuts the axis in T ; prove that 
 
 PQ = 4PT. 
 
 17. Prove that, if the circle of curvature at any point on an ellipse 
 pass through a focus, the point must lie midway between the minor axis 
 and a directrix. 
 
 18. Three points A, P, B are taken on an ellipse whose centre is G. 
 Parallels to the tangents oX A, B drawn from P meet GB and GA 
 respectively in the points Q and R. Shew that Q,R is parallel to the 
 tangent at P. 
 
 19. Two tangents TP, TQ are drawn to an ellipse and any chord 
 TRS is drawn, V being the middle point of the intercepted part; QV 
 meets the ellipse in P' ; prove PP' is parallel to ST. 
 
 20. Shew that the line joining the points of contact of any two 
 parallel tangents, one to each of two given similar and similarly situated 
 ellipses, will pass through one or other of two fixed points. 
 
 21. Shew that, if PP\ DD' and pp', dd' are two pairs of conjugate 
 diameters of an ellipse, the lines P«, Pp' are parallel respectively to 
 D'd, D'd'. 
 
 22. Through the two fixed points A, B ol an ellipse the chords AP, 
 BQ are drawn parallel to each other. Shew that PQ also touches a 
 similar and similarly situated ellipse, 
 
 23. A and B are any two points such that the polar of A with 
 respect to a given ellipse passes through B. From D, the middle point 
 of AB, a tangent DP is drawn to the ellipse. Shew that, if GQ, GR are 
 the semi-diameters parallel to AB and DP, then 
 
 AB : GQ = 2DP : GR. 
 
 4^^ 01 THl 
 
 tliriVBRSITTI 
 
CHAPTER VII. 
 
 Cross-Ratios and Involution. 
 Anharmonic Properties of Conics. 
 
 148. A set of points on a straight line is called a 
 
 range ; and a set of straight lines passing through a point 
 
 is called a pencil, each line being called a ray of the 
 
 pencil. 
 
 If P, Q, R, S be four points on a straight line, the 
 PQ pa 
 
 ratio j^Y) '• nq or PQ . RS : PS . RQ, regard being paid to 
 
 the dwections of the segments [Art. 102], is called the 
 anharmonic ratio or the cross-ratio of the range P, Q, 
 R, S, and is expressed by the notation {PQRS}. 
 
 149. When a straight line PR is divided internally 
 in Q and externally in S in the same ratio, it is said to 
 be divided harmonically; and Q and S are said to 
 be harmonically conjugate with respect to P and R. 
 
 Thus P, Q, R, S is a, harmonic range if 
 
 PQ:QR = P8:RS, 
 
 or PQ:-RQ'=PS:R8. 
 
 Hence the cross of a harmonic range is equal to — 1. 
 When 
 
 {PQP>Sf) = -I, PQ.RS^-RQ.PS; 
 
 ,'. PQ:P8 = PR-PQ:PS-PR, 
 
 so that PQ, PR, PS are in harinonical progression. '^ 
 
CROSS-RATIOS. 205 
 
 If {PQRS}= -1, or PQ.RS=PS .QR and V be the middle 
 point of PR ; then we have 
 
 {PV+VQ) (VS-PV)^iPV+VS) {PV-VQ), 
 whence VQ.VS= VP'- = VEK 
 
 Similarly if W be the middle point of QS 
 WR.WP=WS'^ = WqK 
 
 150. The definition of the cross-ratio of four points 
 on a straight line requires that the points should be taken 
 in a particular order. It follows, however, at once from 
 the definition that 
 
 [PQRS] = {QPSR} = {RSPQ} = {SRQP}. 
 Hence the cross-ratio of four points is unaltered if any 
 two of the points be interchanged and the other two he also 
 interchanged. 
 
 By means of the relation 
 
 PQ . RS + PR . SQ + PS . QR = 0, 
 
 which is true for all positions of P, Q, R, S on a straight line, it can 
 be shewn that if {PQRS} = x, the different values of the cross-ratios, 
 obtained by taking the four points in every possible order, can be 
 shewn to be 
 
 1 , 1 X , l-ac 
 X, - , 1-x, , , = and . 
 
 X 1-a: l-o; x 
 
 151. If a pencil of four straight lines OA, OB, 00, 
 OD be cut by any straight line iyi the points P, Q, R, S 
 respectively ; then will [PQRS] be constant. 
 
206 CROSS-RATIOS. 
 
 Through Q draw XQY parallel to OD and cutting 
 OA, 00 in X, Y respectively. 
 
 Then PQ : PS = XQ : OS, 
 
 and RS:RQ=OS:YQ', 
 
 .'. PQ.RS:PS.ES = XQ:YQ. 
 
 But XQ : YQ is obviously constant for all positions of 
 Q, and hence {PQRS} is constant for all positions and 
 directions of the transversal. 
 
 Def. The cross-ratio of a pencil of four lines OA, 
 OB, 00, OD is the cross-ratio of the range in which the 
 lines are cut by any transversal, and is expressed by the 
 notation {ABGD}. 
 
 Ex. 1. Having given three points on a straight line, find a fourth 
 point on that line such that the range may have a given cross-ratio. 
 
 Let A, By C be the three given points; draw any line AXY 
 through A, and take the points X, Y upon it such that AX : YX is 
 equal to the given cross-ratio. 
 
 Let XB, YG meet in O, and draw through a line parallel to 
 AXY cutting ABC in D. Then D will be the point required. 
 
 For {ABGD) = {AXYcr>]=AX:YX. 
 
 Ex. 2. Having given three lines meeting in a point, find a fourth 
 line through that point such that the pencil may have a given cross- 
 ratio. 
 
 Let OA, OB, OG be the given straight lines. Draw any line 
 cutting OA in Zand OB in F, and take the point Z on this line such 
 
CROSS-RATIOS. 207 
 
 that XY : ZY is equal to the given cross-ratio; Through Z draw a 
 line parallel to OA cutting OG in Z', and let Z'Y cut OA in X\ 
 
 Then the line OD parallel to X'Z' will be the line required. 
 For 0{ABCD} = {X'YZ'cx:>}=X'Y : Z'Y=XY : ZY. 
 
 Ex. 3. If two ranges of equal cross-ratio, on different straight 
 lines, have one common point, the lines joining their other common 
 points will he concurrent. ' 
 
 For, if {ABCI)] = {AB'G'B'], and if BB', CC intersect in 0, and 
 OB cut AB'C'D' in Z/ Then {AB'G'X} = {ABGD} = {AB'G'D'}, 
 whence it follows that X coincides with D'. 
 
 Ex. 4. If two pencils of equal cross-ratio through different points 
 have one common ray, the points of intersection of their other corre- 
 sponding rays are collinear. v 
 
 If {A BGB\ = 0' {AB'G'B'}, and if X, Y, Z be the three points of 
 intersection of the other corresponding rays. Then, if XY does not 
 pass through Z, let it cut 00' in P and OB, O'B' respectively in 
 K,K'. Then 
 
 {ABGB} = {PXYK], and 0' {AB'G'B') = {PXYK'\. 
 
 Thus {PXYK] = {PXYK'}, which is impossible unless K and K' 
 coincide with Z. 
 
 Ex. 5. Each of the three diagonals of a quadrilateral is divided 
 harmonically by the other two diagonals. 
 
 Let the straight lines QAB, QBG, PBA and PGB be the sides of 
 the quadrilateral. The line joining the point of intersection of two 
 of these lines with the point of intersection of the other two is called 
 a diagonal of the quadrilateral. There are therefore three diagonals, 
 viz. PQ, AG, BB, in the figure. 
 
208 
 
 CROSS-RATIOS. 
 
 We have to prove that 
 
 {AOCR} = {BODS} = {QSPR}= -1. 
 
 Let QO cut AD in K and BG in L. 
 
 Then {AOCR} = Q {AOGR} = {AKDP} 
 
 = 0{AKDP} = {GLBP} 
 = Q{GLBP} = {GOAR}. 
 
 And, since {AOGR} = {GOAR 
 
 AO .GR _ GO.AR 
 AR. GO ~ GR.AO ' "' 
 
 {A0GR\=±1. 
 
 We must take the negative sign, for it is easily seen that two of 
 the rays coincide if the anharmonic ratio of a pencil be equal to + 1. 
 
 Hence the diagonal ^C is cut harmonically. 
 
 We can prove in a similar manner that the other diagonals are 
 divided harmonically. 
 
CROSS-RATIOS AND INVOLUTION. 209 
 
 Involution. 
 
 152. Def. When several pairs of points A,A'; B,B'; 
 G, C; &c. lying on a straight line are such that their 
 distances from a fixed point on the line are connected 
 by the relations OA . OA' = OB .OF = 00 . OC = ..., the 
 points are said to form a range in involution^ of which 
 the fixed point is called the centre. 
 
 Two corresponding points, such as A, A', are said to be 
 conjugate to one another. The point conjugate to the 
 centre is at an infinite distance. 
 
 If each point be on the same side of the centre as its 
 conjugate, there will be two points K^, K^y one on each 
 side of the centre, such that 0K^^= 0K.^= OA . OA'. 
 These points K^, K^ are called double points or foci. 
 
 When the points of a conjugate couple are on opposite 
 sides of the centre, the double points are imaginary. 
 
 153. An involution is completely determined when two 
 pairs of conjugate points are given. 
 
 For, if any two circles be drawn through the points 
 A, A' and B, B' respectively, the radical axis of the circles 
 will meet the line A A' BE in a point such that 
 
 OA,OA' = OB,OB\ 
 
 and there is only one such point. 
 
 154. If any number of points he in involution the 
 cross-ratio of any four points is equal to that of their four 
 conjugates. 
 
 Let the pairs of conjugate points be A, A': B, B' \ 
 a, C and D, U. 
 
 Let the radical axis of the circles on ^^' and BB' as 
 diameters cut the line A BCD in the point ; then is 
 the centre of the involution. 
 
 If these circles intersect in real points P, P' ; then 
 A PA', BPB' and also AOP are right angles. 
 
 s. c. 14 
 
210 
 
 CROSS-RATIOS AND INVOLUTION. 
 
 Hence AO.OA' = OP' = BO . OB' = GO . OC = &c., 
 the angles COG\ DOB' are therefore also right angles. 
 
 Hence the angles APB, A'PB' are equal; also the 
 angles BPC, B'PC and the angles GPD, G'PB' are equal. 
 The pencils formed by joining P to the four points 
 A, B, G, D is therefore equiangular to the pencil formed 
 by joining P to the four points A', B', G', D', whence 
 it follows that P{ABGD\ = P{A'B'G'D']. 
 
 If, however, the circles whose diameters are A A' and 
 BB' do not meet in real points, which will be the case when 
 two conjugate points are on the same side of the centre 0, 
 draw a circle through ^, ^' so as to touch the line through 
 perpendicular to AA' ] and let K be the point of 
 contact. 
 
 Then since OK' =OA.OA'=OB. OB', it follows that 
 OK touches in K a circle through B and B', and similarly 
 for the other pairs of points. 
 
CROSS-RATIOS AND INVOLUTION. 
 
 211 
 
 Hence the angles OKA', OAK are equal, and also the 
 angles OKB\ OBK are equal ; therefore the angles 
 A'KB\ AKB are equal. 
 
 Thus the pencil formed by joining K to A, B, G, D\s, 
 equiangular to the pencil formed by joining K to 
 A'y B\ G\ D\ The two pencils are therefore equi-cross. 
 
 It should be noticed that it has been incidentally 
 proved that if two pairs of conjugate points of a range in 
 involution subtend a right angle at any point, every pair 
 will subtend a right angle at that point. 
 
 Cor. I. From the above we obtain a necessary and 
 sufficient condition that three pairs of points may be in 
 involution, namely the condition 
 
 [ABCA']==[A'B'C'A}. 
 
 Cor. II. Any pair of conjugate points in an involution 
 form a harmonic range with the two double points. 
 
 For, if the double points be K^, K^, and A, A' hQ 
 a pair of conjugate points, we have 
 
 {K,AK,A'}=^{K,A'K,A]. 
 
 Ex. 1. Any transversal cuts the three pairs of opposite sides of any 
 quadrangle in three pairs of points in involution. 
 
 Let A, B, G, D he the angular points of the quadrangle. Let 
 AB and CD meet in X, AG and BD in Y, and AD and BG in Z. 
 
 Let any line cut these pairs of opposite sides in P, P' ; Q, Q' and 
 
 14—2 
 
212 CROSS-RATIOS AND INVOLUTION. 
 
 Then {PQBP'}=A {PQRP'} 
 
 = {XGDP'} 
 = B{XCDP'} 
 = {PR'Q'P'} = {P'Q'R'P}. 
 
 Ex. 2. The three pairs of lines from any point to the extremities 
 of the three diagonals of any quadrilateral are in involution. 
 
 155. Def. If any number of pairs of points in invo- 
 lution be joined to any point 0, the pencil so obtained is 
 said to be in involution. 
 
 If A, A'; B, B' ; G, C ; &c. be pairs of points in 
 involution, and if the pencil formed by joining these 
 points to be cut by any other transversal in the pairs of 
 points a, a' ; h,h' \ c, c' ; &c. 
 
 Then since A, A' \ B, B' ; &c. are pairs of points in 
 involution we have relations of the form 
 
 {ABOA'}=:{A'B'C'A}. 
 
 But {ABGA'} = {abca'l 
 
 and {A'B'G'A} = {a'b'c'a}. 
 
 Hence we have 
 
 [abca'} = {a'b'c'a}, &c., 
 
 whence it follows that a, a' ; b,b' ; c, d ; &c. are pairs of 
 points in involution. 
 
 Thus, if a pencil be cut by any transversal in pairs of 
 points in invokttion, it will be cut by any other transversal 
 in pairs of points which are in involution. 
 
 Cor. I. Pairs of perpendicular lines through a point 
 are cut in involution by a straight line. 
 
 Cor. II. Pairs of conjugate diameters of a conic are 
 in involution. 
 
 We know that pairs of conjugate diameters of a conic 
 are cut by any tangent in pairs of points which are 
 in involution, the point of contact of that tangent being 
 the centre of the involution [Art. 108]. 
 
ANHARMONIC PROPERTIES OF CONICS. 
 
 213 
 
 Conjugate diameters are therefore cut by any straight 
 line in pairs of points in involution, the asymptotes of 
 the conic being the double lines of the involution. 
 
 Anharmonic Properties of Conics. 
 
 156. The cross-ratio of the pencil formed by joining 
 any point on a conic to four fixed points is constant and 
 equal to that of the range in which the tangents at those 
 points are cut by any other tangent. 
 
 Let A, B, C, D he four fixed points on a conic whose 
 focus is S, and let P be any other point on the curve. Let 
 PA, PB, PC, PD cut the directrix corresponding to the 
 focus S in a, b, c, d respectively ; and let PS be produced 
 to p. Then we know that Sa will bisect the angle PSA 
 or ASp according as P and A are on opposite branches or 
 on the same branch of the curve. 
 
 From this it follows that, for all positions of P, the 
 angle aSb is constant, being equal or complementary to 
 
214 ANHARMONIC PROPERTIES OF CONICS. 
 
 half the angle A SB according as A and B are on the same 
 or on opposite branches of the curve. Since the angles 
 aSh, hSc, cSd are constant angles, it follows that S [ahcd] 
 is constant. 
 
 But S {ahcd] = {ahcd} 
 
 =^P{ahcd} 
 
 = P{ABGD}. 
 
 Thus A, B, C, D subtend a pencil of constant cross-ratio at 
 any point on the conic. 
 
 Next, let the tangent at P cut the tangents Sit A, B,C,D 
 in a\ h', c\ d' respectively. 
 
 Then we know [Art. 17] that Sa', Sb', Sc\ 8d' are per- 
 pendicular respectively to Sa, 8b, Sc, Sd, and therefore 
 
 S [ab'cd'} = S {ahcd}. 
 
 Hence {a'b'cd'}==S {a'b'c'd'} 
 
 ■^ 8 {abed] =P {abed] 
 
 = P{ABCD]* 
 
 157. The preceding proposition enables us to construct 
 a conic through five given points, or touching five given 
 straight lines. 
 
 If A' be a point on the curve indefinitely near to A ; 
 we have A' {ABCD] = E {ABCD]. 
 
 Thus the cross-ratio of the pencil A' {ABCD] is 
 known ; and A'B, A'G, A'D ultimately coincide with AB, 
 AC, AD respectively. Hence the tangent at A is the line 
 AX through A which is such that A {XBCD] = E {ABCD], 
 We can therefore draw the tangents at the points 
 A, B, CD. 
 
 Then, if the tangents at A and B meet in X, the line 
 through X and F, the middle point of AB, will pass 
 through the centre of the curve. 
 
 Also, if the tangents at B and C intersect in Y, the 
 line through Y and W, the middle point of BC, will pass 
 
 * This proof is due to Mr B. W. Home, late Fellow of St John's College. 
 
ANHARMONIC PROPERTIES OF CONICS. 215 
 
 through the centre. Thus the centre of the conic is 
 determined. 
 
 If be the centre of the conic, and a line through 
 parallel to AB cut the tangent at 5 in T; then, since 
 OX, OT are conjugate diameters, if X, T be on opposite 
 sides of B, the conic must be an ellipse and the square of 
 the diameter conjugate to OB will be equal to XB . BT 
 [Art. 72]; if, however, X and T are on the same side of i?, 
 the conic must be a hyperbola, and the asymptotes will 
 cut the tangent at B in the points L, L' such that 
 
 LB' = BL'^ = BX.BT [Art. 108]. 
 
 In the first case we have a pair of conjugate diameters of 
 an ellipse given in position and magnitude, and the 
 axes &c. can be found as in Art. 77. In the second case 
 we have the asymptotes and a tangent, and the axes &c. 
 can be found as in Art. 114. 
 
 Again, let AB, BC, CD, DE and EA be five given 
 tangents to a conic, and let AB, EA, CD, BC cut DE in 
 the points L, E, D, N respectively, also let AB cut 
 DC in K. 
 
 Then, \i A'B' be a tangent nearly coincident with AB, 
 and if this tangent cuts the tangents AB, EA, CD, BC in 
 the points F, A', K', E respectively, we shall have 
 
 {FA'K'B'} = [LEDN]. 
 
 Now let A'B' move up to and ultimately coincide with 
 AB, then K' will ultimately coincide with K, B' with B, 
 A' with A, and F with the point of contact of the 
 tangent AB. 
 
 Thus [AFBK] = [LEDN], 
 
 and as the cross-ratio of the range A, F, B, K h> known, 
 and three of the points are known, the fourth point which 
 is the point of contact oi AB can be at once found. 
 Having found the points of contact of the five tangents, 
 the construction can be completed as in the former case. 
 
216 ANHARMONIC PROPERTIES OF CONICS. 
 
 From the above constructions it is clear that one conic 
 and only one will pass through five given points, no four 
 of which are on a straight line, and one conic and only 
 one will touch five given straight lines no four of which 
 pass through a point. 
 
 158. The locus of a point which moves so that the 
 pencil formed hy joining it to four fixed points not on a 
 straight line is of constant cross-ratio is a conic through 
 the four given points. 
 
 Let A, B, 0, D hQ the four given points, and let 
 P, Q be any two points such that 
 
 P [ABGD] = Q {ABCD}. 
 
 By the preceding Article one and only one conic will 
 pass through the five points provided no four of the points 
 are on a straight line. 
 
 Hence, if Q be not on the conic determined by the five 
 points A, B, C, D, P, let QA cut this conic in R. 
 
 Then, since R is on the conic through A, B, G, D 
 and P, 
 
 B {ABGD] =:P {ABGD} 
 
 = Q{ABGDl 
 
 whence it follows that B, G and D must be on a straight 
 line ; and similarly it can be shewn that A, B and G are 
 on a straight line. 
 
 Hence Q and R must coincide, for by supposition 
 A, B, G and D are not on a straight line. 
 
 This proves the proposition. 
 
 159. The envelope of a straight line which cuts four 
 fi^ed non-concurrent straight lines in a range of constant 
 cross-ratio is a conic which touches the four fi^ed straight 
 
 Let the four fixed straight lines be cut by two other 
 lines in the points P, Q, R, S and P', Q', R\ S' 
 respectively. 
 
ANHARMONIC PROPERTIES OF CONICS. 217 
 
 Then one conic and only one will touch the four 
 fixed straight lines and the line PQRS ; and, if P'Q'R'8' 
 do not touch this conic, draw another tangent to the conic 
 from 8' and let it cut PF , QQ\ RR in K, L, M 
 respectively. 
 
 Then, by Art. 156, 
 
 [ELMS'] = [PQRB] 
 
 = [P'qR's% 
 
 whence it follows that FKP, Q'LQ, R'MR meet in a 
 point ; and it can be similarly shewn that P'P, Q'Q, S'S 
 will meet in a point. 
 
 Since the four given straight lines do not meet in 
 a point, it follows that the lines P'Q'R'8' and KLM8' 
 must coincide. 
 
 160. If any chord of a conic he drawn through a 
 fixed point 0, it will be cut harmonically by the curve and 
 the polar of 0. 
 
 If be without the conic, the polar of will cut the 
 conic; let the points of intersection be A, B. Let any 
 chord through cut the conic in Q, R and the polar of 
 in F, and let the tangents at Q, R meet in T. 
 
 Then, if A', B' be points on the conic very near 
 io A, B respectively, we have 
 
 A'{AQBR\ = B'[AQBR]. 
 
 Now when A', B' move up to and ultimately coincide 
 with A, B, the above pencils will ultimately be cut by the 
 line OQVQ' in the ranges {OQVR} and {VQOR} respect- 
 ively. 
 
 Hence [OQVR] = {VQOR}, 
 
 whence it follows that QR is divided harmonically in 
 and F. 
 
 Since OT is the polar of the internal point F, the 
 proposition is true for any point, whether external or 
 internal. 
 
218 ANHARMONIC PROPERTIES OF CONICS. 
 
 Conversely, if through any point a line he drawn 
 cutting a conic in the points Q, R, and V he taken on this 
 line such that [OQVE] = - 1 ; then luill V lie on the polar 
 of with respect to the conic. 
 
 161. The cross-ratio of a range of four points on 
 a straight line is equal to that of the pencil formed hy their 
 polars with respect to any conic. 
 
 Let A, B, G, Dhe any four points on a straight line ; 
 then the polars of these points with respect to any conic 
 will all pass through the pole of A BCD with respect 
 to that conic [Art. 115]. 
 
 Let PA\ PB\ PC, PD' be the polars of A, B, C, D 
 with respect to a conic whose focus is S, and let the polars 
 cut the corresponding directrix in the points a, 6, c, d 
 respectively. 
 
 Then, since the angles aSA, hSB, cSG, dSD are 
 all right angles [Art. 17], it follows that 
 
 S\ABGD]=S{ahcd}. 
 
 But S {ahcd} = [ahcd] = P {ahcd} = P [A'BV'D'}. 
 
 Hence {ABGD} = P {A'B'G'D'}. 
 
 162. Def. Two points are said to be conjugate 
 
 points with respect to a conic when each lies on the 
 polar of the other ; also two straight lines are said to be 
 conjugate lines with respect to a conic when each passes 
 through the pole of the other. 
 
 Conjugate diameters of a conic are conjugate lines 
 through the centre of the conic. 
 
 Pairs of conjugate lines with respect to a conic which 
 pass through a point are in involution, the tangents from 
 the point to the conic being the double lines of the invo- 
 lution. 
 
 Let PA, PA' ; PB, PB' ; &c. be any number of pairs 
 of conjugate lines with respect to any conic. 
 
ANHARMONIC PROPERTIES OF CONICS. 219 
 
 Let the polar of P with respect to the conic cut 
 FA', PB\ PG\ PU in the points A\ B\ G\ D' respect- 
 ively ; then A' is the pole of PA, for the pole of PA is by 
 supposition on PA' and it must also be on the polar of P. 
 Thus A', W, C, U are the poles of PA, PB, PC, PD 
 respectively. 
 
 Hence, by the preceding Article, 
 
 P [ABGD] = [A'B'G'U] = P [A'B'G'UY 
 
 whence it follows [Art. 154] that PA, PA' \ PB, PB' ; 
 PG, PG' ; &c. are pairs of lines in involution, and it 
 is obvious that the tangents from P are the double lines 
 of the involution. 
 
 It can also be proved in a similar manner that pairs of 
 conjugate points on a straight line are in involution, the 
 points in which the straight line cuts the conic being the 
 double points of the involution, 
 
 Ex. 1. Through any •point P on a conic the chords PQ, PQ' are 
 drawn making equal angles with the tangent at P. Sheio that QQ' 
 passes through a fixed point. 
 
 Let QQ' cut the tangent at P in the point T, and let the polar of 
 T cut QQ' in F. Then the range T, Q, V, Q' is harmonic, and TP 
 bisects the exterior angle between PQ and PQ', whence it follows that 
 PV bisects the angle QPQ', so that PV is the normal at P. Hence 
 T is a fixed point, namely the pole of the normal chord through P. 
 
 Ex. 2. All conies through four given points have a common 
 self-polar triangle. 
 
 Let A, B, C, D be the four given points [see figure, p. 208]. 
 
 Then, since {BODS}= - 1, it follows that S is on the polar of O 
 with respect to any conic of the system. 
 
 Similarly R is on the polar of with respect to any conic of the 
 system. 
 
 Hence O is the pole of the line SR. 
 
 Again, since {AKDP}= - 1, it follows that K is on the polar of 
 P with respect to any conic of the system. 
 
 Similarly L is on the polar of P. 
 
 Hence P is the pole of the line LOKQ with respect to any conic 
 of the system. 
 
 And, since the polars of and P both pass through Q, Q must be 
 the pole of the line OP. 
 
 Thus the triangle OPQ is such that each angular point is the pole 
 of the opposite side with respect to any conic through the four 
 points A, B, G, D. 
 
220 
 
 ANHARMONIC PROPERTIES OF CONICS. 
 
 Ex. 3. All conies touching four given straight lines have a 
 common self-polar triangle. 
 
 Let AB, BC, CD, DA be the four given straight lines [figure, p. 208]. 
 Let X be the pole of AG with respect to any conic touching the 
 four given straight lines. Then by Art. 160 the pencil 
 
 A{BCDX}=-1 = C{BADX}, 
 
 whence it follows that X must lie on BD, and must coincide with the 
 point S, for we know that {BODS} = - 1. 
 
 Thus S is the pole of ^ C with respect to any conic of the system. 
 
 Similarly R is the pole of BD, and O is the pole of PQ. 
 
 Hence the triangle OSR is such that each angular point is the pole 
 of the opposite loith respect to any conic which touches the four lines 
 AB, BC, CD and DA. 
 
 Since A is on the polar of S, with respect to any conic of the 
 system, the polar of A will go through S, that is the line joining the 
 points of contact of the tangents AB and AD will pass through S ; 
 and similarly for any other pair of the tangents. 
 
 Hence the line joining any two points of contact of any conic of 
 the system will pass through an angular point of the triangle OSR. 
 
 163. Conies through four given points are cut by any 
 straight line in pairs of points in involution. 
 
 Let ABGD be the four given points, and let any 
 straight line cut AG, BD in K, K' respectively, and AD, 
 BC in L, L' respectively. Also let this straight line cut 
 any conic through the four points in P, P' respectively. 
 
 Then A [FDCP}=B[FDGPl since the six points lie 
 on a conic. 
 
ANHARMONIC PROPERTIES OF CONICS. 
 
 221 
 
 Hence taking the ranges formed by these pencils 
 on the line PP\ we have 
 
 [FL'KP] = [FK'LP] ; 
 
 .-. {FL'KP] = {PLK'P% 
 
 which shews that P, P' are conjugate points of the 
 involution determined by the two pairs K, K' and X, L'. 
 
 164. The pairs of tangents drawn from any point to 
 a system of conies touching four given straight lines are in 
 involution. 
 
 Let the four given straight lines be AB, BA\ A'B\ 
 B'A. Let the tangents from to any conic of the system 
 cut ABin the points K, K' and A'B' in the points 2, L'. 
 
 
 Then, since KL, K'L' and the four given straight 
 lines touch the same conic, 
 
 [BKAK'} = {A'LB'L']. 
 
 Hence [BKAK'} = {A'LFL'} 
 
 = 0{A'KB'K]', 
 
 .'. 0[BKAK'} = 0[B'K'A'K], 
 
 which shews that OK, OK' are conjugate rays of the 
 involution determined by the two pairs OA, OA' and 
 OB, OB'. 
 
222 ANHARMONIC PROPERTIES OF CONICS. 
 
 Ex. 1. The locus of the centres of conies ivhich pass through four 
 given points is a conic. 
 
 Let A, B, G, D he the four given points, and let U, V, W, X be 
 the middle points oi AB,BG, CD and DA respectively. Let he the 
 centre of any conic through the four points, and let OP, OQ, OR, 
 OS be parallel to AB, BC, CD, DA respectively. Then OU and OP 
 will be conjugate diameters of the conic, and so also will OV and OQ, 
 OW and OR, and OX and OS. 
 
 Hence, as conjugate diameters are conjugate pairs of an invo- 
 lution, { UVWX} =r O {PQBS}. 
 
 The lines OP, OQ, OR, OS are fixed in direction, and therefore 
 {PQRS} is constant. 
 
 Hence {UVWX\ is constant, and therefore must lie on 
 a fixed conic through U, V, W, X. 
 
 The conic on which the centre lies will also pass through the 
 middle points of ^ C and BD. 
 
 Three conies of the system are the line pairs AB and CD, AC and 
 BD, and AD and BC : the locus of centres must therefore go through 
 the three points of intersection of these pairs of lines. 
 
 Ex. 2. The director-circles of all conies ivhich touch four given 
 straight lines have a common radical axis, and their centres lie 
 on a straight line which passes through the middle points of the 
 diagonals of the quadrilateral formed by the four given straight 
 lines. 
 
 We know [Art. 164] that the tangents drawn to the conies of the 
 system are pairs of lines in involution. If be a point of intersec- 
 tion of any two of the director-circles, two pairs of conjugate rays of 
 a pencil in involution will be at right angles ; every pair will 
 therefore [Art. 154] be at right angles, so that is on the director- 
 circle of every other conic of the system. 
 
 Since every director-circle will pass through the common points 
 of any two of them, the director-circles have a common radical axis, 
 and their centres must therefore lie on a straight line perpendicular 
 to this radical axis. 
 
 Now the double line joining the extremities of a diagonal of the 
 quadrilateral formed by the given straight lines is a limiting form of 
 a conic which touches the lines. Hence the middle point of a 
 diagonal is on the locus of the centres of the conies of the system. 
 The locus of centres must therefore be the straight line through the 
 middle points of the three diagonals of the quadrilateral. 
 
 One of the conies of the system will be a parabola, and the 
 directrix of this parabola will be the common radical axis of the 
 director-circles. 
 
 Ex. 3. Chords of a conic which subtend a right angle at a fixed 
 point on the curve ivill all intersect on the normal at the point. 
 
 Let aa', bb', cc' be three chords of a conic which subtend a right 
 angle at the point on the curve. 
 
ANHARMONIC PROPERTIES OF CONICS. 223 
 
 Then Oa, Oa' ; Oh, Oh' ; Oc, Oc' are pairs of perpendicular lines 
 and are therefore in involution. 
 
 Hence { aba'c ] = { a'h'ac' } . 
 
 But { aha'c } = c' { afta'c } , 
 
 and {a'h'ac'} = h {a'h'ac'). 
 
 Hence c' {aha'c\=h {a'h'axi'] 
 
 These pencils of equal cross-ratio have one ray coincident, namely 
 he' : the intersections of their other corresponding rays must 
 therefore lie on a straight line. Thus a, a' and the point of 
 intersection of cc' and hh' must lie on a line. 
 
 Hence aa', hh' and cc' meet in a point, and therefore every chord 
 which subtends a right angle at must pass through the point of 
 intersection of any two such chords. 
 
 The fixed point through which all the chords pass must lie on the 
 normal at 0, for the normal is a limiting position of one of. 
 the chords. 
 
 Ex. 4. If the perpendicular from a point P on its polar with 
 respect to a given conic pass through a fixed point 0, the point P ivill 
 lie on a rectangular hyperhola whose asymptotes are parallel to the 
 axes of the conic and which passes through the point and through 
 the centre of the conic. 
 
 If the line through P perpendicular to its polar with respect to a 
 conic cut the transverse axis of the conic in G, and PN be the 
 perpendicular on that axis, we know that CG : GN is constant. 
 
 Hence, if PL, PM be parallel to the axes of the conic, the pencil 
 P {GGLM] will be constant. 
 
 But since P { COLM] is constant, it follows that P is on a fixed 
 conic through C, and two points at infinity in the direction of the 
 axes. This proves the proposition. 
 
 A particular case of the above theorem is the following : — 
 
 The points on a conic the normals at tvhich pass through a fixed 
 point O lie on a rectangular hyperhola through and the centre of 
 the conic, and lohose asymptotes are parallel to the axes of the conic. 
 
 Ex. 5. The locus of the pole of a given straight line, loith respect 
 to a system of conies through four fixed points, is a conic. 
 
 Let the given points he A,B, G, D, and let the given line cut AB, 
 BG, GD, DA respectively in P, Q, R, S. Let P' be the point on AB 
 such that {PAP'B} = - 1, and let Q', R', S' be the corresponding 
 points on the other lines. Then the polars of P, Q, R, S with 
 respect to any conic of the system will go through P', Q', R', S' 
 respectively. Hence, if O be the pole of PQRS with respect to any 
 one of the conies, OP, OP', &g. will be pairs of conjugate lines, and 
 therefore {P'Q'R'S'} = {PQRS} = {PQi?S} = constant. 
 
224 
 
 ANHARMONIC PROPERTIES OF CONICS. 
 
 165. PascaPs Theorem. If a hexagcm be inscribed 
 in a conic, the three points of intersection of the three pairs 
 of opposite sides lie on a straight line. 
 
 Let A, B, C, D, E, F be any six points on a conic, and 
 let AB and DE intersect in X, BC and EF in ilf, and CD 
 and FA in N. We have to prove that X, M, N lie on a 
 straight line. 
 
 Let DE cut BG in Z, and DC cut AB in Y. 
 
 Then L [BGDM] = BGXM 
 
 = E{BGXM\ 
 = E[BGDF} 
 = A {BGDF}, 
 
 since the six points 
 are on a conic. 
 
 =={YGDN} 
 = L{YGDN} 
 = L {BGDN], 
 
 whence it follows that L3I and LN are in the same 
 straight line. 
 
 Since six points can be taken in order in sixty 
 different ways, there are sixty hexagons corresponding to 
 
ANHARMONIC PROPERTIES OF CONICS. 225 
 
 six points on a conic ; and, since Pascal's Theorem is true 
 for every one of these hexagons, there are sixty Pascal 
 lines corresponding to six points on a conic. 
 
 166. Brianchon's Theorem. If a hexagon he 
 described about a conic, the three diagonals will meet in 
 a point. 
 
 If a hexagon circumscribe a conic, the points of 
 contact of its sides will be the angular points of a hexagon 
 inscribed in the conic. Each angular point of the circum- 
 scribed hexagon will be the pole of the corresponding side 
 of the inscribed hexagon; therefore a diagonal of the 
 circumscribed hexagon, that is a line joining a pair of its 
 opposite angular points, will be the polar of the point of 
 intersection of a pair of opposite sides of the inscribed 
 hexagon. But the three points of intersection of pairs of 
 opposite sides of the inscribed hexagon lie on a straight 
 line by Pascal's Theorem ; hence their three polars, that 
 is the three diagonals of the circumscribed hexagon, will 
 meet in a point. 
 
 If we are given five tangents to a conic, the points of contact of 
 the tangents can be found by Brianchon's Theorem. 
 
 For, let A, B, G, D, E be the angular points of a pentagon 
 formed by the five given tangents, and let K be the point of contact 
 of AB ; then A, K, B, G, D, E will be the angular points of 
 a circumscribing hexagon, two sides of which are coincident. By 
 Brianchon's Theorem, DK passes through the point of intersection 
 of AC and BE, hence K is found, and the other points of contact can 
 be found in a similar manner. 
 
 We can also find the tangents to a conic at five given points on 
 the curve by means of Pascal's Theorem. 
 
 For, let A, B, G, D, E be the five given points, and let F be the 
 point on the conic indefinitely near to A ; then, by Pascal's Theorem, 
 the three points of intersection of AB and DE ; of BG and EF ; and 
 of GD and FA lie on a straight line. Hence, if the line joining the 
 point of intersection of AB and DE to the point of intersection of J5C 
 and EA meet GD in H, then AH will be the tangent at A. The 
 other points of intersection can be found in a similar manner. 
 
 167. The following examples are important. 
 
 Ex. 1. If tioo triangles circumscribe a conic their six angular 
 points lie on another conic. 
 
 Let ABG, A'B'G' be the two triangles. 
 
 S, c, 15 
 
226 ANHARMONIC PROPERTIES OF CONICS. 
 
 Let B'C cut AB, AC in E', D' respectively ; and let BG cut A'B', 
 A'C in E, B respectively. 
 
 Then the tangents BG, B'G' will cut the remaining four tangents 
 in ranges of equal cross-ratio. 
 
 Hence {BGEB} = {E'D'B'G'} \ 
 
 :. A' {BGEB]=A{E'B'B'G']\ 
 
 i.e. A' {BGB'G']=A {BGB'G'}, 
 
 which proves the proposition. 
 
 We can now prove that, if one triangle can he inscribed in a given 
 conic and circumscribed to another given conic, an infinite number of 
 triangles can be so described. 
 
 For, let ABC he a triangle inscribed in the conic S and circum- 
 scribed about the conic S'. Draw any tangent to S' and let it cut S 
 in the points B', C. Let the other tangents to S' from B' and 
 G' meet in A'. 
 
 Then, we have proved that A', B', G' , A, B, G lie on a conic, and 
 five of these points lie on the conic S ; the sixth point must therefore 
 lie on S, for only one conic will go through five given points. 
 
 Ex. 2. If two triangles be inscribed in a conic, their six sides 
 touch a second conic. 
 
 Let the triangles he ABC and A' B'C. 
 
 Let BG cut A'B', A'C' in E, F respectively, and let B'C' cut AB, 
 AG in E', F' respectively. 
 
 Then, since the six points A, B, G, A', B', G' lie on a conic, 
 
 A{BGB'C'}=A'{BGB'G'\; 
 
 .-. {E'F'B'G'} = {BCEF\, 
 
 which proves the proposition. 
 
 Ex. 3. If two triangles be self-polar with respect to any conid 
 their six angular points lie on a second conic, and their six sides touch 
 a third conic. 
 
 Let ABC, A'B'G' be two triangles which are self-polar with 
 respect to any conic. 
 
 Let A'B', A'C' cut BG in K, L respectively. 
 
 Then, since B' is the pole of A'C and A the pole of BG, AB' will 
 be the polar of L. Similarly AC will be the polar of K. 
 
 Now the pencil formed by any four lines through a point and the 
 range formed by their poles are of equal cross-ratio. 
 
 Hence A {BGB'G'} = {GBLK} 
 
 = A'{GBLK} 
 
 = A'{GBC'B'\ 
 
 = A'{BGB'G'}, 
 
 whence it follows that A, B, G, A', B', C lie on a coniq, 
 
ANHARMONIC PROPEETIES OF CONICS. 227 
 
 Again, if B'C cut AB, AG in F, G respectively, we have 
 {GBLK}=A{BGB'G'} 
 = {FGB'G'} 
 = {GFG'B'}, 
 so that BG and B'G' cut the other four sides in ranges of equal 
 cross-ratio, whence it follows that the six sides touch the same conic. 
 We can now prove that, if one triangle can be inscribed in 
 {or circumscribed about) one given conic and self-polar with respect 
 to another given conic, an infinite number of triangles can be so 
 described. [See Ex. 1.] 
 
 168. Homographic ranges and pencils. Ranges 
 and pencils are said to be homographic when every four 
 constituents of the one, and the corresponding four 
 constituents of the other, have equal cross-ratios. 
 
 Ex. 1. The points of intersection of corresponding lines of tioo 
 homographic pencils describe a conic. 
 
 Let P, Q, R, S be any four of the points of intersection of corre- 
 sponding lines, and 0, 0' the vertices of the pencils. 
 
 Then, by supposition, {PQRS] = 0' {PQRS}, whence it follows 
 that 0, 0', P, Q, P, S lie on a conic. But five points are sufficient 
 to determine a conic ; hence the conic through 0, O' and any three 
 of the intersections will pass through every other intersection. 
 
 Ex. 2. The lines joining corresponding points of two homographic 
 ranges on different straight lines envelope a conic. 
 
 Ex. 3. AB, A'B' are any two finite lines ; shew that if P, P' are 
 points on these lines respectively such that AP : PB=A'P' : P'B', the 
 line PP' will envelope a parabola. 
 
 Ex, 4. Find the common lines of tivo homographic pencils through 
 the same point. 
 
 Let OA, OB, OG be any three rays of one pencil and OA', OB' 
 OG' the three corresponding rays of the other. Then we have to find* 
 the line OP which is such that {ABGP]=0 {A'B'G'P}. 
 
 Draw any circle through and let the points A, B, G, A', B', G' 
 lie on this circle. 
 
 Let AB', A'B meet in X and AG', A'G in Y ; and let XY cut the 
 circle at a point P. 
 
 Then, if AA' cut the line XYP in Z, we have 
 
 {ZXYP]=A {ZXYP]=A {A' B'G'P] = {A'B'G'P}, 
 since the points are all on the circle ; and similarly 
 {ZXYP}=:0{ABGP}. 
 
 Hence OP is one of the required straight lines ; and from the 
 construction it will be seen that any two homographic pencils will 
 have two real, coincident, or imaginary common rays. 
 
 15—2 
 
228 ANHARMONIC PROPERTIES OF CONICS. 
 
 Ex. 5. Find the common points of two homographic ranges on the 
 same straight line. 
 
 Join the points to any point 0, and proceed as Ex. in 4. 
 
 Ex. 6. The three sides of a triangle pass through fixed points, and 
 the extremities of its base lie on two fixed straight lines ; shew that its 
 vertex describes a conic. 
 
 Let A, B, C be the three fixed points, and let Oa, Oa' be the two 
 fixed straight lines. Suppose triangles drawn as in the figure. 
 
 Then the ranges {abed...} and {a'b'c'd'...} are homographic. 
 Therefore the pencils B {abed...} and G {a'b'c'd'...} are homographic, 
 and the result follows from Ex. 1. 
 
 O a b 
 
 The above is MacLaurin's method of generating a conic. 
 
 Ex. 7. If all the sides of a polygon pass through fixed points, and 
 all the angular points but one move on fixed straight lines ; the remain- 
 ing angular point will describe a conic. 
 
 169. Circular points at infinity. Since any pair 
 of perpendicular lines through the centre of a circle are 
 conjugate, and since pairs of conjugate lines with respect 
 to a conic which pass through any point are in involution, 
 the real or imaginary tangents from that point to the 
 conic being the double lines of the involution, it follows 
 that the imaginary asymptotes of all circles are parallel, 
 so that all circles go through the same two imaginary 
 points at infinity ; also concentric circles have common 
 
ANHAKMONIC PROPERTIES OF CONICS. 229 
 
 imaginary asymptotes and therefore have double contact 
 with one another at infinity. 
 
 The two imaginary points at infinity through which 
 all circles pass are called the circular points at 
 infinity. 
 
 Again, since any pair of perpendicular lines through a 
 focus of a conic are conjugate, and since these pairs 
 of .conjugate lines are in involution of which the imaginary 
 tangents from the focus are the double lines, it follows that 
 the imaginary tangents to a conic from a focus are parallel 
 to the imaginary asymptotes of any circle, so that the 
 tangents to any conic through a focus pass through the 
 circular points at infinity. 
 
 Thus all conies with one focus common have two 
 common imaginary tangents through that focus, and 
 confocal conies have four imaginary common tangents. 
 
 170. We have shewn how to construct the conic 
 which passes through five given points or which touches 
 five given straight lines. The following other cases are of 
 interest. 
 
 Ex. 1. To construct a conic which passes through four given 
 points and touches a given straight line. 
 
 Let the given straight line cut two pairs of opposite sides of the 
 quadrangle formed by the four given points in A, A' and jB, B'. 
 Then, by Desargue's Theorem, all conies through the four given 
 points are cut in pairs of conjugate points of the involution deter- 
 mined by the pairs A, A' and B, B' ; hence if a conic through the four 
 points touch the given line, the point of contact must be one of the 
 double points of the involution. Thus there are two (real or 
 imaginary) conies which pass through four given points and touch a 
 given straight line ; and since five points on either conic are known 
 the construction can be completed as in Art. 157. 
 
 Ex. 2. To construct a conic which touches four given straight lines 
 and passes through a given point. 
 
 The tangent at the given point can be found by means of the 
 reciprocal of Desargue's Theorem. [Art. 164.] 
 
 Ex. 3. To consti-uct a conic passing through three given points 
 and touching two given straight lines. 
 
 Let AB, JC be the given straight lines, and D, E, F the given 
 points. 
 
230 ANHARMONIC PROPERTIES OF CONICS. 
 
 Let any conic through D, E touch AB, AG in L, M respectively, 
 and let DE cut LM in the point P, and A B, AG in the points £', C" 
 respectively. 
 
 Then, the lines AB, AG, the double line LMP, and the conic 
 through D, E are three conies through the same four points, namely 
 two coincident points at each of the points L, M. These conies are 
 therefore cut in involution by the line DE, and therefore P is 
 one of the double points of the involution determined by the pairs 
 D, E and B', G'. Thus the chord of contact of the tangents AB, AG 
 passes through one or other of two fixed points on I)E. Similarly the 
 chord of contact passes through one or other of two fixed points on IDF. 
 
 The chord of contact is therefore one of four fixed straight lines, 
 . and if either of these lines cut AB, AC in the points X, Y, there is one 
 and only one corresponding conic through the five points B, E, F, 
 X, Y, which can be constructed as in Art. 157 or Art. 166. 
 
 Thus four conies will pass through three given points and touch 
 two given straight lines. 
 
 Ex. 4. To construct a conic passing through tioo given points and 
 touching three given straight lines. 
 
 Let ABO be the triangle formed by the three given tangents, and 
 let D, E be the two given points. 
 
 Then, as in Ex. 3, the chord of contact of the tangents from A 
 passes through one or other of two fixed points, S, S' suppose, 
 on DE. 
 
 If AX be the polar of S with respect to the conic A {SEXD} = - 1, 
 and therefore AX can be constructed; and AX', the polar of S', can 
 be similarly constructed. Hence 0, the pole of DE, is on one 
 or other of two fixed lines through A. 
 
 Similarly is on one or other of two fixed lines through B. 
 
 Hence is one of four fixed points ; and when is found, OD 
 and OE will be the corresponding tangents to the conic, which 
 is now completely determined since five tangents are known. 
 
 Thus four conies will pass through two given points and touch 
 three given straight lines. 
 
 Ex. 5. Gonstruct a conic having given the poles loith respect to it 
 of three given points. 
 
 Let BG, GA, AB be the polars of the points A', B', G' 
 respectively. 
 
 Then, if K be the point of intersection of lines through B', G' 
 parallel to AB, GA respectively, the centre of the conic will lie on the 
 line AK. And, if L be the point of intersection of lines through A', 
 C' parallel to AB, BG respectively, the centre of the conic will lie on 
 the line BL [Art. 125]. Hence the centre, 0, of the conic is found. 
 
 Let OA', OB', OG' cut BG, GA, AB in the points P, Q, R respect- 
 ively. If A' and P be on the same side of 0, and X be such that 
 OX^=zOA' . OP, the hne through X parallel to BG will be a tangent. 
 Hence, if the conic be an ellipse, the tangents at three points can be 
 found ; and if the tangent at the point X be cut by OP', and a line 
 
ANHARMONIC PROPERTIES OF CONICS. 231 
 
 through parallel to CA in the points T, T respectively, the rect- 
 angle TX . XT' will be equal to the square of the semi-diameter 
 conjugate to OX. Thus a pair of conjugate diameters of the ellipse 
 are found in position and magnitude, and the construction can 
 be completed as in Art. 77. 
 
 If the conic be a hyperbola the asymptotes will be the double lines 
 of the pencil in involution of which OA' and the line through 
 O parallel to BG are jone conjugate pair, and OB' and the line 
 through parallel to CA' are another conjugate pair ; and knowing 
 the asymptotes and the pole of a given straight line the construction 
 can be completed. 
 
 171. The rectangle contained by the two perpendiculars 
 drawn from any point of a conic on one pair of opposite 
 sides of an inscribed quadrilateral, is in a constant ratio to 
 the rectangle contained by the two perpendiculars from that 
 point on another pair of opposite sides. [Pappus's 
 Theorem.] 
 
 Let A, B, G, D he four given points on a conic, and 
 let be any other point on the curve. Let Oa, Ob, Oc, 
 Od be the perpendiculars from on AB, BG, GD, DA 
 respectively. 
 
 Then we have to prove that the ratio Oa. Oc : Ob. Od 
 is constant for all positions of on the conic. 
 
 Let OB, OD cut ^C in the points X, F respectively. 
 
 Then {ABGD} = [AXGY] = AX . YG : XG .AY. 
 
 lS^owAX:XG=AOAB: AOBG=Oa.AB : Ob.BG; 
 
 and YG:AY=ADGO:AOAD=OG.GD'.Od.DA. 
 
 Hence, as {ABGD} is constant for all positions of 
 on the conic, it follows that Oa . Oc : Ob . Od is constant. 
 
 EXAMPLES. 
 
 1. Having given five points on a conic, shew how to find any 
 number of other points on the curve. 
 
 2. Having given five tangents to a conic, shew how to draw any 
 number of other tangents to the curve. 
 
 3. Construct a conic, having given the centre and three points. 
 
 4. Construct a conic, having given the centre and three tangents. 
 
232 EXAMPLES. 
 
 5. Construct a conic, having given two points on the curve and a 
 triangle which is self-polar with respect to it. 
 
 6. Construct a conic, having given two tangents and a triangle 
 which is self-polar with respect to the conic. 
 
 7. Find the centre of a rectangular hyperbola which touches four 
 given straight lines. 
 
 8. Shew that the locus of the centres of rectangular hyperbolas with 
 respect to which a given triangle is self-polar is the circle circumscribing 
 the triangle. 
 
 9. Shew that, if a triangle be inscribed in the triangle ABC and 
 touch the side BG on the point F, the centre of the conic will lie on the 
 straight line through the middle points of BC and AF. 
 
 10. A, B, G, D are any four points on a hyperbola ; GK parallel to 
 one asymptote meets AD in K and DL parallel to the other asymptote 
 meets GB in L. Shew that KL is parallel to AB. 
 
 11. Shew that the sixty Pascal lines corresponding to six points on 
 a conic, intersect three by three. 
 
 12. The poles with respect to a conic of the sides BG, GA, AB are 
 A', B', G' respectively. Shew that AA', BB', GG' meet in a point, and 
 that the points of intersection of BG and B'G', GA and G'A', AB and A'B' 
 are coUinear. 
 
 13. Find geometrically the points where a given straight line cuts 
 the conic determined by five given points. 
 
 14. Find geometrically the tangents from a given point to the conic 
 determined by five given tangents. 
 
 15. Through a fixed point on a conic a line is drawn cutting the 
 conic again in P and the sides of a given inscribed triangle in A\ B', G' 
 respectively. Shew that {PA'B'G') is constant. 
 
 16. On a fixed tangent to a conic any point is taken and OQ is 
 the other tangent from to the conic. Shew that, if ABG be the 
 vertices of any triangle circumscribing the conic, 0{ABGQ\ will be 
 constant for all positions of O. 
 
 17. Two points P and Q are conjugate with respect to a conic, P 
 lies on a fixed straight line and QP subtends a right angle at a fixed 
 point. Prove that the locus of Q is a conic passing through the fixed 
 point. 
 
 18. From a fixed point on one of the three diagonals of a 
 complete quadrilateral tangents are drawn to the conies inscribed in the 
 quadrilateral, shew that their points of contact lie on a conic which 
 passes through the extremities of the two other diagonals and divides 
 harmonically the diagonal on which O lies. 
 
 19. A straight line cuts two given circles in points which are 
 harmonically conjugate. Shew that the line envelopes a conic whose 
 foci are at the centres of the circles. 
 
EXAMPLES. 233 
 
 20. Shew how, by the method of false positions or otherwise, to 
 describe a polygon each of whose sides shall pass through a fixed point, 
 and each of whose vertices shall lie on a fixed straight line. 
 
 21. Shew that, if three conies pass through the same four points, a 
 common tangent to any two of the conies is cut harmonically by the 
 third. 
 
 22. Shew that, if three conies touch the same four straight lines, 
 the tangents to two of the conies at a common point and the two tangents 
 drawn from that point to the other conic form a harmonic pencil. 
 
 23. A hyperbola passes through the centre of a conic and its 
 asymptotes are parallel to a pair of conjugate diameters of the conic ; 
 shew that an infinite number of triangles can be inscribed in the 
 hyperbola which are self-polar with respect to the conic. 
 
 24. Shew that, if ABGDEF be any hexagon inscribed in a conic, 
 the continued product of the perpendiculars drawn from any point on 
 the conic to the sides AB, CD and EF is in a constant ratio to the 
 continued product of the perpendiculars drawn from the same point on 
 the alternate sides BG, DE and FA. 
 
 25. Shew that, if a conic touch the four straight lines AB^ BG, GD, 
 DA, and any other tangent to the conic cut AD, BG in P, Q respectively; 
 then will AP : PB and BQ : GQ be in a constant ratio. Hence, or 
 otherwise, shew that the rectangle contained by the perpendiculars from 
 A and G on any other tangent to the conic is in a constant ratio to the 
 rectangle contained by the perpendicular from B and D on that tangent. 
 
 26. Q is any point on a given straight line, R is the point of 
 intersection of the polars of Q with respect to two given conies. Shew 
 that the locus of R for different positions of Q is a conic. 
 
 27. Any straight line is drawn through a given point 0, and Q, Q' 
 are the poles of the line with respect to two given conies ; shew that the 
 envelope of QQ', for different straight lines through O, is a conic. 
 
 28. From any two points T, T' the tangents TP, TQ and T'P', T'Q' 
 are drawn to a conic ; shew that the six points T, T', P, P', Q, Q' lie on 
 another conic. 
 
 29. Shew that the circum-circle of a triangle self-polar to a conic 
 cuts orthogonally the director-circle of the conic. 
 
 30. Shew that the locus of the centres of conies inscribed in a 
 triangle and whose director- circles have a given radius is a circle. 
 
 31. Shew that the locus of the centres of the rectangular hyperbolas 
 which touch the sides of a triangle is the polar circle of the triangle. 
 
 32. Shew that the director-circles of all conies which touch the 
 sides of a triangle cut orthogonally the polar circle of the triangle. 
 
CHAPTER VIII. 
 Reciprocation. Projection. 
 
 172. If we have any figure consisting of any number 
 of points and straight lines in a plane, and we take 
 the polars of those points and the poles of the line with 
 respect to a fixed conic G, we shall obtain another figure 
 which is called the polar reciprocal of the former with 
 respect to the auxiliary conic C. 
 
 When a point in one figure and a line in the reciprocal 
 figure are pole and polar with respect to the auxiliary 
 conic C, they are said to correspond to one another. 
 
 If in one figure we have any curve S the lines which 
 correspond to the different points of S will all touch some 
 curve S\ Let the lines corresponding to the two points 
 P, Q oi S meet in T ; then T will be the pole of the line 
 PQ with respect to the conic 0, that is the line PQ 
 corresponds to the point T. Now if the point Q move up 
 to and ultimately coincide with P, the two corresponding 
 tangents to S' will also ultimately coincide with one 
 another, and their point of intersection T will ultimately 
 be on the curve S' and will coincide with the point 
 of contact of the line which corresponds to the point P. 
 So that a tangent to the curve S corresponds to a point on 
 the curve S', just as a tangent to S' corresponds to a 
 point on S. Thus S is generated from S' exactly as S^ is 
 from 8, and we shall arrive at the same curve S' either as 
 
RECIPROCATION. 235 
 
 the envelope of the polars of the different points on 8 or 
 as the locus of the poles of the different tangents to S. 
 
 173. If any line L cut the curve S in any number 
 of points P, Q, R, ... we shall have tangents to S' corre- 
 sponding to the points P, Q, R, ..., and these tangents 
 will all pass through a point, namely through the pole of 
 L with respect to the auxiliary conic. Hence as many 
 tangents to S' can be drawn through a point as there are 
 points on S lying on a straight line ; so also as many 
 points on S' will lie on a straight line as these tangents to 
 8 drawn through a point. 
 
 The number of points, real or imaginary, in which 
 a curve is cut by any straight line is called the degree 
 of the curve, and the number of tangents which can 
 be drawn to a curve from any point is called the class 
 of the curve. 
 
 Thus the degree of a curve is equal to the class of its 
 reciprocal, and the class of a curve is equal to the degree 
 of its reciprocal. 
 
 We know that a conic is of the second degree and 
 of the second class ; hence the reciprocal of a conic is also 
 of the second degree and of the second class. We cannot 
 however conclude from this result that the reciprocal of a 
 conic is another conic, unless we prove that no other 
 curves except conies are of the second degree and of the 
 second class. 
 
 174. The polar reciprocal of one conic with respect to 
 another is a conic. 
 
 Let A, B, G, DhQ any four fixed points, and let their 
 polars with respect to the auxiliary conic be A'A'\ B'B", 
 G'C", UD". 
 
 Let P be any other point and let P'P" be its polar, 
 and let P'P" cut.^'^", BIB', CO", D'D" in the points 
 a, b, c, d respectively. 
 
 Then, since a is the intersection of the polars of A and 
 
 UHIVBESITY] 
 
236 RECIPROCATION. 
 
 P, a is the pole of the line PA. Similarly h, c, d are the 
 poles of the lines PB, PG, PD respectively. 
 
 Hence P [ABGD] = {abed}. 
 
 Now if P be any point on a fixed conic through 
 A, B, C, D we know that P [ABGD] is constant. 
 
 Hence [abed] is constant, which shews that the line 
 abed touches a fixed conic which touches the lines A'A'\ 
 B'B'\ G'G", D'D". 
 
 175. The method of Reciprocal Polars enables us to 
 obtain from any given theorem concerning the positions 
 of points and lines, another theorem in which straight 
 lines take the place of points and points of straight 
 lines. 
 
 The simplest cases of correspondence are the follow- 
 ing:— 
 
 Points in one figure reciprocate ihto straight lines in the reciprocal 
 figure. 
 
 The line joining two points reciprocate into the point of inter- 
 section of the corresponding lines. 
 
 The tangent to any curve reciprocates into a point on the 
 reciprocal curve. 
 
 The point of contact of a tangent reciprocates into the tangent at 
 the corresponding point. 
 
 The chord joining any two points on a curve reciprocates into 
 the intersection of the corresponding tangents to the reciprocal 
 curve. 
 
 The tangents to any curve from the centre of the auxiliary conic 
 reciprocate into the points at infinity on the reciprocal curve. 
 
 The points of contact of the tangents drawn to any curve from 
 the centre of the auxiliary conic reciprocate into the tangents at the 
 points at infinity, that is into the asymptotes r of the reciprocal 
 curve, 
 
 [Hence, if the original curve be a conic, the line joining the 
 points of contact of the tangents drawn from the centre of the 
 auxiliary conic reciprocates into the centre of the reciprocal conic ; 
 also the reciprocal of a conic will be an ellipse, parabola, or 
 hyperbola according as the tangents drawn to it from the centre 
 of the auxiliary conic are imaginary, coincident, or real; that is 
 according as the centre of the auxiliary conic is within, upon, or 
 without the original conic] 
 
 If two curves touch, that is have two coincident points common, 
 the reciprocal curves will have two coincident tangents common, 
 and will therefore also touch. 
 
RECIPROCATION. 
 
 237 
 
 176. The following are examples of reciprocal 
 theorems. 
 
 If the angular points of two 
 triangles are on a conic, their six 
 sides will touch another conic. 
 
 The three intersections of oppo- 
 site sides of a hexagon inscribed in 
 a conic lie on a straight line. 
 
 (Pascal's Theorem.) 
 
 If the three sides of a triangle 
 touch a conic, and two of its angu- 
 lar points lie on a second conic, the 
 locus of the third angular point is 
 a conic. 
 
 If the sides of a triangle touch 
 a conic, the three lines joining an 
 angular point to the point of con- 
 tact of the opposite side meet in a 
 point. 
 
 The polars of a given point with 
 respect to a system of conies through 
 four given points all pass through 
 a fixed point. 
 
 The locus of the pole of a given 
 line with respect to a system of 
 conies through four fixed points is 
 a conic. 
 
 If the sides of two triangles 
 touch a conic, their six angular 
 points are on another conic. 
 
 The three lines joining opposite 
 angular points of a hexagon de- 
 scribed about a conic meet in a 
 point. (Brianchon's Theorem.) 
 
 If the three angular points of a 
 triangle lie on a conic, and two of 
 its sides touch a second conic, the 
 envelope of the third side is a 
 conic. 
 
 If the angular points of a tri- 
 angle lie on a conic, the three points 
 of intersection of a side and the 
 tangent at the opposite angular 
 point lie on a line. 
 
 The poles of a given straight 
 line with respect to a system of 
 conies touching four given straight 
 lines all lie on a fixed straight line. 
 
 The envelope of the polar of a 
 given point with respect to a system 
 of conies touching four fixed lines 
 is a conic. 
 
 177. We now proceed to consider the results which 
 can be obtained by reciprocating with respect to a 
 circle. 
 
 We know that the line joining the centre of a circle to 
 any point P is perpendicular to the polar of P with respect 
 to the circle. Hence, if P, Q be any two points, the angle 
 between the polars of these points with respect to a circle 
 is equal to the angle that PQ subtends at the centre of 
 the circle. Reciprocally the angle between any two 
 straight lines is equal to the angle which the line joining 
 their poles with respect to a circle subtends at the centre 
 of the circle. 
 
 We know also that the distances, from the centre of 
 a circle, of any point and of its polar with respect to that 
 circle, are inversely proportional to one another. 
 
238 EECIPROCATION. 
 
 178. If we reciprocate with respect to a circle it is 
 clear that a change in the radius of the auxiliary circle 
 will make no change in the shape of the reciprocal curve, 
 but only in its size. Hence, as we are generally not 
 concerned with the absolute magnitudes of the lines in 
 the reciprocal figure, we only require to know the centre of 
 the auxiliary circle. We may therefore speak of recipro- 
 cating with respect to a point 0, instead of with respect to 
 any circle having for centre. 
 
 179. , If any conic be reciprocated with respect to 
 a point 0, the points on the reciprocal figure which 
 correspond to the tangents through to the original 
 curve must be at an infinite distance in directions 
 perpendicular to those tangents. Thus the directions of 
 the lines to the points at infinity on the reciprocal curve 
 are perpendicular to the tangents from to the original 
 curve ; and hence the angle between the asymptotes of the 
 reciprocal curve is supplementary to the angle between the 
 tangents from^ to the original curve. 
 
 In particular, if the tangents from to the original 
 curve be at right angles, the reciprocal conic will be a 
 rectangular hyperbola. 
 
 Again, the axes of the reciprocal conic bisect the 
 angles between its asymptotes. The axes are therefore 
 parallel to the bisectors of the angles between the tangents 
 from to the original conic. 
 
 Corresponding to the points at infinity on the original 
 conic we have the tangents to the reciprocal conic which 
 pass through the origin. • Hence the tangents from the 
 origin to the reciprocal conic are perpendicular to the 
 directions of the lines to the points at infinity on the 
 original conic, so that the angle between the asymptotes of 
 the original conic is supplementary to the angle between 
 the tangents from the origin to the reciprocal conic. 
 
 In particular, if a rectangular hyperbola be recipro- 
 cated with respect to any point 0, the tangents from to 
 the reciprocal conic will be at right angles to one another ; 
 
RECIPROCATION. 239 
 
 in other words is a point on the director-circle of the 
 reciprocal conic. 
 
 180. Let any line through cut a given conic in the 
 points P, P\ and let the tangents at P, P' meet in T\ 
 then we know that T is on the polar of with respect to 
 the given conic. 
 
 If now we reciprocate with respect to : — correspond- 
 ing to the points P, P' on a line through are two 
 parallel tangents to the reciprocal conic, and corresponding 
 to the intersection of the tangents to the original conic at 
 P, P' is the line joining the points of contact of these 
 parallel tangents. Hence corresponding to the line on 
 which T lies is the point through which the chord of 
 contact of parallel tangents to the reciprocal conic passes, 
 and this point is the centre of the reciprocal conic. 
 
 Thus the polar of the origin with respect to any conic 
 reciprocates into the centre of the reciprocal conic. 
 
 181. The following are important examples of recipro- 
 cation : 
 
 We know that all conies which circumscribe a triangle 
 and pass through its orthocentre are rectangular hyper- 
 bolas. 
 
 Reciprocating with respect to the orthocentre we 
 shall obtain another triangle with the same orthocentre. 
 
 The rectangular hyperbolas will become parabolas, 
 since they all pass through 0; and, since the points 
 at infinity on any one of the conies are in perpendicular 
 directions, the tangents from to any one of the parabolas 
 will be at right angles, so that the point is on the 
 directrix of each parabola. 
 
 Thus the reciprocal theorem is the directrices of all 
 parabolas which touch the three sides of a given triangle 
 pass through the orthocentre of the triangle. 
 
 Again, the original theorem may be expressed in the 
 form : — - 
 
240 
 
 KECIPKOCATION. 
 
 182. If two of the conies which pass through four 
 given points are rectangular hyperbolas, they will all be 
 rectangular hyperbolas. 
 
 If this be reciprocated with respect to any point 0, we 
 shall obtain the following theorem. 
 
 If the tangents from a point to two of the conies 
 which touch four given straight lines be at right angles to 
 one another, the tangents from to any conic of the 
 system will be at right angles to one another. Hence all 
 the director-circles of conies touching four given straight 
 lines will pass through either of the points of intersection 
 of any two of them. 
 
 Thus the director-circles of all conies which touch four 
 given straight lines have a common radical axis. 
 
 183. To find the polar reciprocal of one circle with 
 respect to another. 
 
 Let be the centre of the given circle to be recipro- 
 cated, and >Si the centre of the auxiliary circle. 
 
 Draw any line through S cutting the given circle 
 in the points P, P\ 
 
RECIPROCATION. 241 
 
 Upon SP take the point Y such that SP . SY may be 
 equal to the square of the radius of the auxiliary circle. 
 Then the line through Y perpendicular to SP will be the 
 polar of P with respect to the auxiliary circle, and will 
 therefore be a tangent to the reciprocal curve. 
 
 Since the rectangles SP . 8Y and SP.8P' are both 
 constant, it follows that the ratio SY : SP' is constant, and 
 therefore the locus of Y is similar to the locus of P' ; the 
 locus of Y for different positions of P is therefore a circle. 
 [If the point S be on the circumference of the given circle, 
 the locus of Y will be a straight line, that is a circle 
 of infinite radius.] 
 
 But a straight line which moves so that the locus of 
 the foot of the perpendicular upon it, drawn from a fixed 
 point, is a circle, envelopes a conic of which the fixed 
 point is a focus. 
 
 Hence the polar reciprocal of a circle with respect 
 to any point S is a conic of which S is a focus. 
 
 Draw YG parallel to OP' and cutting OS in F. 
 Then, from similar triangles, 
 
 SG:SO = SY:SF = constant, 
 
 and CYiCS =0P' :0S= constant. 
 
 Thus G is the centre of the reciprocal conic, and GY is 
 the length of its semi-major- axis. 
 
 Hence the eccentricity of the conies is equal to 
 
 GS: GY=OS: OP'. 
 
 The reciprocal conic is therefore an ellipse, a parabola, 
 or a hyper'hola according as the point S is within, upon, or 
 without the given circle. 
 
 Let OD be the radius of the circle which is perpen- 
 dicular to OS. Draw SN, the perpendicular on the 
 tangent at D, and take a point L on SN such that SN . SL 
 may be equal to the square of the radius of the auxiliary 
 circle. Then L will be the point on the reciprocal conic 
 which corresponds to the tangent DN, and SL will be the 
 
 s. c. 16 
 
242 RECIPROCATION. 
 
 semi-latus-rectum of the conic. And, since SN = OD, it 
 follows that the diameter of the auxiliary circle is a mean 
 proportional to the diameter of the given circle and the 
 latus-rectum of the reciprocal conic. 
 
 Again, since L is the polar of DN with respect to the 
 auxiliary circle, the polar of D will pass through Z, and it 
 will be perpendicular to 8D ; hence, as this polar touches 
 the reciprocal conic, it must be the tangent at the point 
 L. Thus the tangent at L is the line through L perpen- 
 dicular to 8D; and this tangent will cut /SfO in a point X 
 which is the foot of the directrix corresponding to the 
 focus ;S^. 
 
 Now, since the angles DOX and DFX are right 
 angles, 
 
 SO ,SX = SD. SF 
 
 = square of radius of auxiliary circle, 
 
 whence it follows that the centre of the given circle 
 reciprocates into the directrix of the conic. 
 
 If the polar of 8 with respect to the given circle 
 cut OS in the point K, 08 .0K= OP', and therefore 
 
 08 . SK= 0P'-0S'=^P8 . SP'. 
 
 Hence SK : SP = SP' : OS 
 
 = 8Y:SG; 
 
 .-. SK .8G = SP.SY, 
 
 whence it follows that the polar of 8 with respect to 
 the given circle reciprocates into the centre of the reciprocal 
 conic. [See Art. 180.] 
 
 Ex. 1. Tangents to a conic subtend equal angles at a focus. 
 
 Reciprocate with respect to the focus : — then corresponding to the 
 two tangents to the conic, there are two points on a circle ; the point 
 of intersection of the tangents to the conic corresponds to the line 
 joining the two points on the circle ; and the points of contact of the 
 tangents to the conic correspond to the tangents at the points on the 
 circle. Also the angle subtended at the focus of the conic by any 
 two points is equal to the angle between the lines corresponding to 
 
RECIPROCATION. '*-2i3_ 
 
 those two points. Hence the reciprocal theorem is — The line joining 
 two points on a circle makes equal angles with the tangents at 
 tiiose points. 
 
 Ex. 2. The envelope of the chord of a conic which subtends 
 a right angle at a fixed point is a conic having for a focus, and 
 the polar of 0, ^oith respect to the original conic, for the corresponding 
 directrix. 
 '--^ Reciprocate with respect to 0, and the proposition becomes — The 
 locus of the point of intersection of tangents to a conic which are at 
 right angles to one another is a concentric circle. 
 
 Ex. 3. If two conies have a common focus, two of their common 
 chords will pass through the intersection of their directrices. 
 
 Reciprocate with respect to the common focus, and the pro- 
 position becomes-'-Two of the points of intersection of the common 
 tangents to two circles are on the line joining the centres of the 
 circles. 
 
 — ' Ex. 4. The orthocentre of a triangle circumscribing a parabola is 
 on the directrix. 
 
 Reciprocating with respect to the orthocentre we obtain — A 
 conic circumscribing a triangle and passing through the orthocentre 
 is a rectangular hyperbola. 
 
 184. To reciprocate a system of circles with the same 
 radical axis into a system of confocal conies. 
 
 If we reciprocate with respect to any point we 
 obtain a system of conies having for one focus, and 
 [Art. 183] the centre of any conic is the reciprocal of the 
 polar of with respect to the corresponding circle. Now 
 either of the two ' limiting points ' of the system is such 
 that its polar with respect to any circle of the system is 
 a fixed straight line, namely a line through the other 
 limiting point parallel to the radical axis. If therefore the 
 system of circles be reciprocated with respect to a limiting 
 point the reciprocals will have the same centre; and if 
 they have a common centre and one common focus they 
 will be confocal. Since the radical axis is parallel to and 
 midway between a limiting point and its polar, the re- 
 ciprocal of the radical axis (with respect to the limiting 
 point) is on the line through the focus and centre of the 
 reciprocal conies, and is twice as far from the focus as the 
 
 16—2 
 
244 CONICAL PROJECTION. 
 
 centre ; so that when we reciprocate a system of coaxial 
 circles with respect to a limiting point, the radical axis 
 reciprocates into the other focus of the system of confocal 
 conies. 
 
 The following theorems are reciprocal : 
 
 The tangents at a common The points of contact of a 
 
 point of two confocal conies are at common tangent to two circles 
 
 right angles. subtend a right angle at one of the 
 
 limiting points. 
 
 The locus of the point of inter- The envelope of the line joining 
 
 section of two lines, each of which two points, each of which is on one 
 
 touches one of two confocal conies, of two circles, and which subtend 
 
 and which are at right angles to a right angle at a limiting point, 
 
 one another, is a circle. is a conic one of whose foci is at 
 
 the limiting point. 
 
 If from any point two pairs of If any straight line cut two 
 
 tangents P, P' and Q, Q' be drawn circles in the points P, P' and 
 
 to two confocal conies ; the angle Q, Q' ; the angles subtended at a 
 
 between P and Q is equal to that limiting point by PQ and P'Q' are 
 
 between P' and Q'. equal. 
 
 Conical Projection. 
 
 185. If any point P be joined to a fixed point V, and 
 VF be cut by any fixed plane in P', the point P' is called 
 the projection of P on that plane. The point V is called 
 the vertex or the centre of projection, and the cutting plane 
 is called the plane of projection. 
 
 186. The projection of any straight line is a straight 
 line. 
 
 For the straight lines joining V to all the points of 
 any straight line are in a plane, and this is cut by the 
 plane of projection in a straight line. 
 
 187. Any plane curve is projected into a curve of the 
 same degree. 
 
 For, if any straight line meet the original curve in 
 any number of points A, B, G, D..., the projection of the 
 line will meet the projection of the curve where VA, VB, 
 
CONICAL PROJECTION. 245 
 
 VG, VD... meet the plane of projection. There will 
 therefore be the same number of points on a straight 
 line in the one curve as in the other. This proves the 
 proposition. 
 
 188. A tangent to a curve projects into a tangent 
 to the projected curve. 
 
 For, if a straight line meet a curve in two points A, B, 
 the projection of that line will meet the projected curve 
 in two points a, h where VA, VB meet the plane of 
 projection. Now if A and B coincide, so also will a 
 and b. 
 
 189. The relation of pole and polar with respect to a 
 conic are unaltered by projection. 
 
 This follows from the two preceding Articles. 
 
 It is also clear that two conjugate points, or two 
 conjugate lines, with respect to a conic, project into 
 conjugate points, or lines, with respect to the projected 
 conic. 
 
 190. Draw through the vertex a plane parallel to the 
 plane of projection, and let it cut the original plane in the 
 line K'L\ Then, since the plane VK'L' and the plane of 
 projection are parallel, their line of intersection, which is 
 the projection of K'L\ is at an infinite distance. 
 
 Hence to project any particular straight line K'L' to 
 an infinite distance, take any point V for vertex and 
 a plane parallel to the plane VK'U for the plane of 
 projection. 
 
 Straight lines which meet in any point on the line 
 K'L' will be projected into parallel straight lines, for their 
 point of intersection will be projected to infinity. 
 
 191. A system of parallel lines on the original plane 
 will be projected into lines which meet in a point. 
 
 For, let VP be the line through the vertex parallel 
 to the system, P being on the plane of projection; then, 
 
246 
 
 CONICAL PROJECTION. 
 
 since FP is in the plane through V and any one of the 
 parallel lines, the projection of every one of the parallel 
 lines will pass through P. 
 
 For different systems of parallel lines the point P will 
 change ; but, since VP is always parallel to the original 
 plane, the point P is alw^ays on the line of intersection of 
 the plane of projection and a plane through the vertex 
 parallel to the original plane. 
 
 Hence any system of parallel lines on the original 
 plane is projected into a system of lines passing through 
 a point, and all such points, for different systems of 
 parallel lines, are on a straight line. 
 
 192. Let KL be the line of intersection of the original 
 plane and the plane of projection. Draw through the 
 vertex a plane parallel to the plane of projection, and let 
 it cut the original plane in the line K'L\ Let the two 
 straight lines AOA', BOB' meet the lines KL, K'L in 
 the points A, B and A', B' respectively ; and let VO meet 
 the plane of projection in 0'. Then AO' and BO' are the 
 projections oi AOA' and BOB'. 
 
CONICAL PROJECTION. 247 
 
 Since the planes VA'B\ A O'B are parallel, and parallel 
 planes are cut by the same plane in parallel lines, the lines 
 VA\ VB' are parallel respectively to AO', BO'. The angle 
 A'VB' is therefore equal to the angle ^0'jB,that is, A'VB' 
 is equal to the angle into which AOB is projected. 
 
 Similarly, if the straight lines CD, ED, meet K'L' in 
 C, D' respectively, the angle C'VD' will be equal to the 
 angle into which ODE is projected. 
 
 From the above we obtain the fundamental proposition 
 in the theory of projections, viz., 
 
 A7iy straight line can be projected to infinity, and at the 
 same time any two angles into given angles. 
 
 For, let the straight lines bounding the two angles meet 
 the line which is to be projected to infinity in the points 
 A\ B' and C, U ; draw any plane through A'B'G'D', and 
 in that plane draw segments of circles through A', B' and 
 G\ D' respectively containing angles equal to the two given 
 angles. Either of the points of intersection of these 
 segments of circles may be taken for the centre of pro- 
 jection, and the plane of projection must be taken parallel 
 to the plane we have drawn through A'B'G'D'. 
 
 If the segments do not meet, the centre of projection 
 is imaginary. 
 
 Ex. 1. To shew that any quadrilateral can be projected into 
 a square. 
 
 Let ABGD be the quadrilateral ; and let P, Q [see figure p. 
 208] be the points of intersection of a pair of opposite sides, and 
 let the diagonals BD, AG meet the line PQ in the points S, B. 
 Then, if we project PQ to infinity and at the same time the angles 
 PDQ and ROS into right angles, the projection must be a square. 
 For, since PQ is projected to infinity, the pairs of opposite sides of 
 the projection will be parallel, that is to say, the projection is a 
 parallelogram ; also one of the angles of the parallelogram is a 
 right angle, and the angle between the diagonals is a right angle j 
 hence the projection is a square. 
 
 Ex. 2. To shew that the triangle formed by the diagonals of 
 a quadrilateral is self-polar with respect to any conic which touches 
 the sides of the quadrilateral. 
 
 Project the quadrilateral into a square; then, the circle 
 circumscribing the square is the director-circle of the conic, therefore 
 
248 CONICAL PROJECTION. 
 
 the intersection of the diagonals of the square is the centre of 
 the conic. 
 
 Now the polar of the centre is the line at infinity ; hence the 
 polar of the point of intersection of two of the diagonals is the 
 third diagonal. 
 
 Ex. 3. If a conic he inscribed in a quadrilateral the line joining 
 two of the points of contact will pass through one of the angular 
 points of the triangle formed by the diagonals of the quadrilateral. 
 
 Ex. 4. If ABC be a triangle circumscribing a parabola, and the 
 parallelograms ABA'G, BCB'A, and GAC'B be completed; then the 
 chords of contact will pass respectively through A', B\ C. 
 
 This is a particular case of Ex. 3, one side of the quadrilateral 
 being the line at infinity. 
 
 Ex. 5. If the three lines joining the angular points of two 
 triangles meet in a point, the three points of intersection of correspond- 
 ing sides icill lie on a straight line. 
 
 Project two of the points of intersection of corresponding sides 
 to infinity, then two pairs of corresponding sides will be parallel, 
 and it is easy to shew that the third pair will also be parallel. 
 
 Ex. 6. Any two conies can be projected into concentric conies. 
 [See Art. 162, Ex. 2.] 
 
 193. Any conic can be projected into a circle having 
 the projection of any given point for centre. 
 
 Let be the point whose projection is to be the 
 centre of the projected curve. 
 
 Let P be any point on the polar of 0, and let OQ be 
 the polar of P ; then OP and OQ are conjugate lines. 
 
 Take OP', OQ' another pair of conjugate lines. 
 
 Then project the polar of to infinity, and the angles 
 
CONICAL PROJECTION. 249 
 
 POQ, P'OQ' into right angles. We shall then have a 
 conic whose centre is the projection of 0, and since two 
 pairs of conjugate diameters are at right angles, the conic 
 is a circle. 
 
 194. A system of conies inscribed in a quadrilateral 
 can he projected into confocal conies. 
 
 Let two of the sides of the quadrilateral intersect 
 in the point A, and the other two in the point B. Draw 
 any conic through the points A, B, and project this conic 
 into a circle, the line AB being projected to infinity ; then, 
 A, B are projected into the circular points at infinity, and 
 since the tangents from the circular points at infinity to 
 all the conies of the system are the same, the conies must 
 be confocal. 
 
 Ex. 1. Conies through four given points ean be projected into 
 coaxial circles. 
 
 For, project the line joining two of the points to infinity, and one 
 of the conies into a circle ; then all the conies will be projected into 
 circles, for they all go through the circular points at infinity. 
 
 Ex. 2. Conies which have double contact with one another can be 
 projected into concentric circles. 
 
 Ex. 3. The three points of intersection of opposite sides of 
 a hexagon inscribed in a conic lie on a straight line. [Pascal's 
 Theorem.] 
 
 Project the conic into a circle, and the line joining the points of 
 intersection of two pairs of opposite sides to infinity ; then we have 
 to prove that if two pairs of opposite sides of a hexagon inscribed in 
 a circle are parallel, the third pair are also parallel. 
 
 Ex. 4. Shew that all conies through four fixed points can be 
 projected into rectangular hyperbolas. 
 
 There are three pairs of lines through the four points, and 
 if two of the angles between these pairs of lines be projected into 
 right angles, all the conies will be projected into rectangular 
 hyperbolas. [Art. 121.] 
 
 Ex. 5. If two triangles are self-polar with respect to a conic, 
 their six angular points are on a conic, and their six sides tou^h a 
 conic. 
 
 Let the triangles be ABC, A'B'C Project EC to infinity, and 
 the conic into a circle ; then A is projected into the centre of the 
 
250 CONICAL PROJECTION. 
 
 circle, and AB, AC are at right angles, since ABC is self-polar ; also, 
 since A'B'C is self -polar with respect to the circle, A is the ortho- 
 centre of the triangle A'B'C 
 
 Now a rectangular hyperbola through A', B', C will pass 
 through A, and a rectangular hyperbola through B will go through C. 
 Hence, since a rectangular hyperbola can be drawn through any four 
 points, the six points A, B, C, A', B\ C are on a conic. 
 
 Also a parabola can be drawn to touch the four straight lines 
 B'C, C'A', A'B', AB. And A is on the directrix of the parabola 
 [Art. 53, Ex. 4] ; therefore ylC is a tangent. Hence a conic touches 
 the six sides of the two triangles. 
 
 195. Properties of a figure which are true for any 
 projection of that figure are called projective properties. 
 In general such properties do not involve magnitudes. 
 There are however some projective properties in which 
 the magnitudes of lines and angles are involved : the 
 most important of these is the following: — 
 
 The cross-ratios of pencils and ranges are unaltered 
 by projection. 
 
 Let A, B, C, D he four points in a straight line, and 
 A', B\ C, n be their projections. Then, if V be the 
 centre of projection, VAA', VBB\ VGG\ VDU are straight 
 lines; and we have [Art. 151] 
 
 [ABGD] = V[ABGD] = {A'B'G'D']. 
 
 If we have any pencil of four straight lines meeting in 
 0, and these be cut by any transversal 'm. A,B,G,D\ then 
 
 [ABGD] = [ABGD] = V [ABGD] = [A'B'G'B'} 
 
 = 0' {A'B'G'D'}. 
 
 From the above together with Article 154 it follows that 
 if any number of points be in involution, their projections 
 will be in involution. 
 
 Ex. 1. Any chord of a conic through a given "point O is divided 
 harmonically by the curve and the polar of 0. 
 
 Project the polar of to infinity, then is the centre of the 
 projection, the chord therefore is bisected in 0, and {POQco } 
 is harmonic when PO = OQ. 
 
CONICAL PKOJECTION. 251 
 
 Ex. 2. Conies through four fixed points are cut by any straight 
 line in pairs of points in involution. [Desargue's Theorem.] 
 
 Project two of the points into the circular points at infinity, then 
 the conies are projected into coaxial circles, and the proposition is 
 obvious. 
 
 Ex. 3. If AOA', BOB', COG', DOB',... he chords of a conic, the 
 points A, B, C, B,... and the points A', B', C, D',... will subtend 
 homographic pencils at any point on the curve. 
 
 Project the conic into a circle having O for centre. 
 
 Ex. 4. If there are two systems of points on a conic which 
 subtend homographic pencils at any point on the curve, the lines joining 
 corresponding points of the two systems will envelope a conic having 
 double contact icith the original conic. 
 
 Let A, B, G, D,... and A', B', C, D',... be the two systems 
 of points. 
 
 Let AB', A'B meet in K, and AC, A'C in L; and project 
 the conic into a circle, KL being projected to infinity. Then, it is 
 easily seen that AA', BB' and CC will be projected into equal chords 
 of the circle ; and therefore the angles subtended by AB and BG &t 
 any point on the circle will be equal respectively to the angles 
 subtended by A'B' and B'C'. 
 
 Hence, if P, P' be any other pair of corresponding points, and 
 be any point on the circle, since O {ABCP) = O {A'B'G'P'}, it follows 
 that the angles COP and COP' are equal, and that 
 
 PP' = GC = BB'=AA'. 
 
 The envelope of PP' is therefore a concentric circle. 
 
 Ex. 5. In a given conic inscribe a triangle each of whose sides 
 will pass through a given fixed point. 
 
 Let P, Q, R be the three fixed points through which the sides 
 pass. 
 
 Draw any chord B'C through the point P, and let CQ cut the 
 curve again in A', and let A'R cut the curve again in D'. 
 
 Let any other points B", B'" be taken on the conic, and the 
 corresponding points B", D'" be found. 
 
 Let X be either of the points on the conic which are such that 
 
 {B'B"B"'X} = {D'D"D"'X}, 
 
 where is any point on the conic [see Art. 168, Ex. 4]. Then, if XP 
 cut the curve again in Y, and YQ cut the curve in Z, ZX will pass 
 through E, and XYZ will be one of the two real or imaginary triangles 
 which satisfy the required conditions. 
 
252 CONICAL PROJECTION. 
 
 196. The following are additional examples of the 
 methods of reciprocation and projection. 
 
 Ex. 1. If the sides of a triangle touch a conic, and if tioo of the 
 angular points move on fixed confocal conies, the third angular point 
 loill describe a confocal conic. 
 
 Let ABC, A'B'C be two indefinitely near positions of the triangle, 
 and let AA', BB', GC produced form the triangle PQR. The six 
 points A, By C, A', B', C are on a conic [Art. 167, Ex. 1], and this 
 conic will ultimately touch the sides of PQR in the points A, B, G. 
 Hence PA, QB, EG will meet in a point; and it is easily seen that 
 the pencils A { QGPB], B [RAQG], G {PBRA } are harmonic. Now, 
 if A move on a conic confocal to that which AB, AG touch, the 
 tangent at A, that is the line QR, will make equal angles with AB, 
 AG. Hence, since A {QGPB} is harmonic, PA is perpendicular to 
 QR. Similarly, if B move on a confocal, QB is perpendicular to 
 RP. Hence RG must be perpendicular to PQ, and therefore GA, GB 
 make equal angles with PQ ; whence it follows that G moves on a 
 confocal conic. 
 
 [The proposition can easily be extended. For, let ABGB be a 
 quadrilateral circumscribing a conic, and let A, B, G move on 
 confocals. Let DA, GB meet in E, and AB, DG in F. Then, by 
 considering the triangles ABE, BGF, we see that E and F move on 
 confocals. Hence, by considering the triangle GED, we see that D 
 will move on a confocal.] 
 
 If we reciprocate with respect to a focus we obtain the following 
 theorem : 
 
 If the angular points of a triangle are on a circle of a co-axial 
 system, and two of the sides touch circles of the system, the third side 
 will touch another circle of the system. [Poncelet's theorem.] 
 
 Ex. 2. The six lines joining the angular points of a triangle to the 
 points where the opposite sides are cut by a conic, will touch another 
 conic. 
 
 The reciprocal theorem is : — 
 
 The six points of intersection of the sides of a triangle with the 
 tangents to a conic draion from the opposite angular points , will lie on 
 another conic. 
 
 Project two of the points into the circular points at infinity, then 
 the opposite angular point of the triangle will be projected into a 
 focus, and we have the obvious theorem : — 
 
 Two lines through a focus of a conic are cut by pairs of tangents 
 parallel to them in four points on a circle. 
 
 Ex. 3. The following theorems are deducible from one another. 
 
 (i) Two lines at right angles to one another are tangents one to 
 each of two confocal conies ; sheio that the locus of their intersection 
 is a circle, and that the envelope of the line joining their points of 
 contact is another confocal. 
 
EXAMPLES. 253 
 
 (ii) Two points, one on each of two co-axial circles, subtend a 
 right angle at a limiting point ; shew that the envelope of the line 
 joining them is a conic with one focus at the limiting point, and that 
 the locus of the intersection of the tangents at the points is a co-axial 
 circle. 
 
 (iii) Two lines which are tangents one to each of two conies, cut a 
 diagonal of their circumscribing quadrilateral harmonically ; sheio that 
 the locus of the intersection of the lines is a conic through the extremi- 
 ties of that diagonal, and that the envelope of the line joining the 
 points of contact is a conic inscribed in the same quadrilateral. 
 
 (iv) AOB, COD are common chords of two conies, and P, Q are 
 points, one on each conic, such that {APBQ} is harmonic ; shew that 
 the envelope of the line PQ is a conic touching AB, CD, and that the 
 tangents at P, Q meet on a conic through A, B, G, D. 
 
 (v) If two points be taken, one on each of two circles, equidistant 
 from their radical axis, the envelope of the line joining them is a 
 parabola which touches the radical axis, and the locus of the inter- 
 section of the tangents at the points is a circle through their common 
 points. 
 
 EXAMPLES. 
 
 1. Shew that four conies can be described having a common focus 
 and passing through three given points, and that the latus-rectum of one 
 of the conies is equal to the sum of the latera recta of the other three. 
 
 2. Reciprocate with respect to any point the theorem : — ' The 
 tangents to a circle from any point make equal angles with their chord 
 of contact.' 
 
 3. Reciprocate the following : — * The common chord of two circles 
 is perpendicular to the line joining the centres of the circles.' 
 
 4. Reciprocate the following : — ' Four circles will touch three given 
 straight lines, and the line joining the centres of any two of the circles 
 will pass through one of the points of intersection of the given straight 
 lines.' 
 
 5. Shew that, if a triangle be reciprocated with respect to any point 
 on its circum-circle, the point will be on the circum-circle of the 
 reciprocal triangle, 
 
 6. Two parabolas touch the sides of a given triangle and cut one 
 another at right angles at P. Shew that the point P must lie on the 
 circum-circle of the given triangle. 
 
 7. Reciprocate with respect to B the following theorem: — 'If any 
 one of a system of circles through the two fixed points A, B he cut by 
 any given straight line through A in the point P, the tangent at P will, 
 for different circles of the system, envelope a parabola whose focus is B,' 
 
254 EXAMPLES. 
 
 8. Eeciprocate with respect to one of the limiting points the 
 theorem: — 'The tangents drawn to all the circles of a coaxial system 
 from any point on the radical axis are equal. ' 
 
 9. Eeciprocate with respect to and 0' the theorem : — * The locus 
 of the centre of a circle which passes through a fixed point and 
 touches a fixed circle whose centre is 0' is a conic whose foci are and 
 0'.' 
 
 10. Eeciprocate and so prove the following theorem : — * Two chords 
 of a rectangular hyperbola are at right angles to one another, and each 
 subtends a right angle at a fixed point P. ' Shew that the locus of their 
 intersection is the polar of P. 
 
 11. Any line is drawn through a fixed point cutting a given conic 
 in the points P, Q, and the point R is taken on the line such that 
 {OPQR} is constant. Shew that the locus of J? is a conic having double 
 contact with the given conic. 
 
 12. Shew that two circles and their centres of similitude subtend a 
 pencil in involution at any point. 
 
 13. Shew that, if P, P' be corresponding points of two homcgrapLic 
 ranges on the fixed lines OA, OA' respectively, and the parallelogram 
 POP'Q be completed, the locus of Q will be a conic. 
 
 14. Shew that any conic which passes through the three fixed 
 points A, B, G, and is such that two other given points are conjugate 
 with respect to it, will pass through another fixed point. 
 
 15. Shew that the locus of the centre of a conic which passes 
 through the two fixed points A, B, and has also two given pairs of 
 conjugate points, is a conic. / 
 
 16. A conic circumscribes a triangle and its director-circle passes 
 through the orthocentre of the triangle; shew that the polar of the 
 orthocentre with respect to the conic touches the polar-circle of the 
 triangle. 
 
 17. Prove that, if a conic be drawn through the four points of 
 intersection of two given conies, and through the intersection of one 
 pair of common tangents, it also passes through the intersection of jthe 
 other pair of common tangents. 
 
 18. Prove that the locus of the vertex from which a system of four 
 fixed points in a plane can be projected into a square is a circle in a 
 plane at right angles to the third diagonal of the quadrilateral formed by 
 the four points. 
 
 19. Shew that any two triangles in plane perspective can be 
 projected into equilateral triangles. 
 
EXAMPLES. 255 
 
 20. A. circle and a rectangular hyperbola are described each with its 
 centre on the other curve ; a parabola is described with its focus at the 
 centre of the hyperbola and its directrix touching the hyperbola at the 
 centre of the circle ; prove that there are an infinite number of triangles, 
 which are at the same time inscribed in one of the three curves, 
 circumscribed about another and self-polar with respect to the third, in 
 any order. 
 
 21. Shew that the envelope of the axes of conies which touch two 
 given straight lines at fixed points is a parabola. 
 
 22. Shew that if one conic inscribed in a quadrilateral is a circle, 
 the axes of any other inscribed conic are the asymptotes of a rectangular 
 hyperbola which circumscribes the diagonal triangle of the quadrilateral ; 
 shew also that the axes envelope a parabola which touches the diagonals 
 of the quadrilateral and whose directrix passes through the middle 
 points of the diagonals. 
 
 flHIVBRSIIT] 
 
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