UC-NRLF $B 525 ^2fl GEOMETRICAL CONIC SECTIONS IN MEMORIAL Irving Stringham i Jt ¥ A GEOMETRICAL TREATISE ox CONIC SECTIONS. ( JEOMETRICAL TREATISE ON CONIC SECTIONS. WITH NUMEROUS EXAMPLES. Jfor fyz Wist of JScbooIs anb S&iabjente in the (Unibrrsxtics. WITH AN APPENDIX ON HARMONIC RATIO, POLES AND POLARS, AND RECIPROCATION. BY THE KEY. W, H. DEEW, M.A., // st. John's college, Cambridge, professor ok mathematics in king's college, london. SEVENTH EDITION. bonbon : MACMILLAN AND CO. 1 883 . \The Right of Translation and Reproduction is Reserved.'] I . 4 K. Clay, Sons, and Taylotj, BREAD STREET HILL, E.C. ^5£ 1?^ PREFACE TO THE FIFTH EDITION. In this Edition an Appendix has been added, in which an endeavour has been made to present the subject of Harmonic Ratio, Poles and Polars, and Reciprocation, in a form adapted to the wants of Students who approach these ideas for the first time. A further collection of Problems has also been given, taken from Examination Papers of recent dates, and the work has thus been brought completely up to the requirements of the present time. W. H. DREW. King's College, London, June 26, 1875. 800571. CONTENTS, introduction PAGB 1 CHAPTER I. THE PARABOLA '•$ PROBLKMS ON IMF. PARABOLA 25 CHAPTER II. THE ELLIPSE 29 PROBLKMS ON Till'. ELLIPSE 63 CHAPTER III. THE HYPERBOLA 69 PROBLEMS ON THE HYPERBOLA H»7 CHAPTER IV. THE SECTIONS OF THE CONK 11- PROP.LEMS ON THE SECTIONS OF THE CONK 123 ADDITIONAL PROBLEMS 124 SECOND SERIFS 139 APPENDIX 1^3 CONIC SECTIONS. INTRODUCTION. 1. Def. The curve traced out by a point, which moves in such a manner that its distance from a given fixed point continually bears the same ratio to its distance from a given fixed line, is called a Conic Section. The fixed point is called the Focus, and the fixed line the Directrix. Thus if S be the focus, and KK' the directrix, and P a point from which PM is drawn at right angles to the directrix, the curve traced out by P ^ will be a Conic Section, provided P move in such a manner that SP always bears the same ratio to PM. (1.) "When the distance from the fixed point is equal to the distance from the fixed line, that is, when SP is equal to PM, the Conic Section is called a Parabola. K' (2.) When the distance from the fixed point is less than the distance from the fixed line, that is, when the ratio which 2 B 2 CONIC SECTIONS. SP bears to I'M is less than unity, the Conic Section is called an Ellipse. (3.) When the distance from the fixed point is greater than the distance from the fixed line, that is, when the ratio which SP bears to PM is greater than unity, the Conic Section is called an Hyperbola. 2. The reason of the term Conic Sections being applied to these curves is that, when a Cone is intersected by a plane surface, the boundary of the section so formed will, in general, be one or other of these curves. I propose to investigate the properties of the Conic Sections from the definitions given above, and afterwards to show in what manner a Cone must be divided by a plane in order that the curve of intersection may be a Parabola, Ellipse, or Hyperbola. CHAPTEE I. THE PARABOLA. Prop. I. 3. The focus and directrix of a parabola being given, to find any number of points on the curve. Let 8 be the focus, and KK' the directrix. Draw X Sx at right angles to the directrix, and bisect the line SX in A ; then since AS — AX, . \ A is a point on the curve. The point A is called the Vertex, and the line Ax, with respect to which the curve is evidently symmetrical, is called the Axis. 4 CONIC SECTIONS., On the directrix take any point M ; join SM ; and draw MP at right angles to the directrix. At the focus S make the angle MSP equal to the angle SMP; then SP = PM, .'. Pis' a point on the curve. So by taking any number of points, M ', M" , on the directrix, we may obtain as many points, P', P", on the curve as we please, and the line which passes through A and all these points will be the parabola whose focus is S and directrix KK'. Cor. 1. As M is taken further away from the point X, the line SM and the angles SMP, MSP, and, consequently, the lines SP and PM, continually increase. Hence, since XM and MP increase together, the curve recedes at the same time both from the axis and directrix ; and since the angle SMP can never exceed a right angle, and the lines $Pand MP will therefore always meet, it is evident that there is no limit to the distance to which the curve may extend on both sides of the axis. Cor. 2. The parabola may be described practically in the m K following manner M =3<2 A' :» s x Let S be the focus and A~A"be the directrix; and let a rigid bar QM, having a string of the same length as itself fastened at one end Q, be made to slide parallel to the axis with the other end M on the directrix ; then if the other end of the string be fastened at the focus, and the string be kept stretched by means of the point of a pencil at P, in contact with the bar, since SP will always be equal to PM, it is evident that the point P will trace out the parabola. CONIC SECTIONS. o Prop. II. 4. The distance of any point inside the parabola from the focus is less than its distance from the directrix; and the distance of any point outside the parabola from the focus is greater than its distance from the directrix. (1.) Let Q be a point inside the parabola. Draw QM at right angles to the directrix, meeting the parabola in P ; join SP ; then since SP = PM, .■.SPan&PQ = QM. But SP and PQ > SQ, .\QM>SQ. (2.) Let Q be a point outside the parabola. Draw MQ at right angles to the directrix, and produce it to meet the parabola in P ; join SP ; then since SQ and QP > SP, and SP = PM, .-. SQ and QP > PM, .-.SQ>QM. Cor. Conversely a point will be inside or outside the parabola according as its distance from the focus is less or greater than its distance from the directrix. 5. Dee. The line PN {see fig. Prop. III.) drawn at right angles to the axis from the point P in the curve is called the Ordinate of the point P, and the line AN the Abscissa. The double ordinate B C drawn through the focus, and termi- nated both ways by the curve, is called the Latus Pedum. CONIC SECTIONS. Prop. III. The Latus Eectum BO = 4: AS. Draw BK at right angles to the directrix. Then SB = BK = SX = 2 AS, .\BC=4AS 6. Def. If a point P' be taken on the parabola (see fig. Prop. IV.) near to P, and PP' be joined, the line PP' pro- duced, in the limiting position which it assumes when P' is made to approach indefinitely near to P, is called the Tangent to the parabola at the point P. Prop. IV. If the tangent to the parabola at any point P intersect the directrix in the point Z ; then SZ will be at right angles to SP. Let P r be a point on the parabola near to P. CONIC SECTIONS. Draw the chord PP' and produce it to meet the directrix in Z; join SZ. Draw PM, P' M' at right angles to the directrix ; join SP, SP' ; and produce PS to meet the parahola in Q. Then, since the triangles ZMP, ZM' P' are similar, ..ZPxZP' v.MPiM ' P\ ::SP: SP', .'. SZ bisects the angle P'SQ. {Euclid, VI. Prop. A.) Now when P' is indefinitely near to P, and PP' becomes the tangent at the point P, the angle PSP' becomes in- definitely small, while the angle QSP' approaches two right angles, and therefore the angle P' SZ, which is half of the angle P' SQ, becomes ultimately a right angle. Hence, when PZ is the tangent, the anc^le ZSP is a rmht anode, or SZ is perpendicular to SP. Cor. Conversely, if SZ be drawn at right angles to SP, meeting the directrix in Z, and PZ be joined, P Z will be a tangent at P. 3 CONIC SECTIONS. Prop. Y. 7. The taDgent at any point P of a parabola bisects the angle between the focal distance SP, and the perpendicular PM on the directrix. E Let the tangent at P meet the directrix in the point Z ; join SZ; then since the angle ZSP is a right angle, (Prop. IV.) .-. ZS 2 + SP 2 = PZ 2 . Also ZAP + IIP 2 = PZ 2 . .-. ZS 2 4 SP 2 = ZM 2 + MP 2 . But SP = PM, ,.ZS=ZM. Now in the triangles ZPS, ZPM, .-. ZP,PS= ZP, PM, each to each, and ZS = ZM, . ' . the angle SPZ = the angle MPZ ; or PZ bisects the angle SPM. CONIC SECTIONS. 9 Cor. 1. If ZP be produced to R, then the angle SPR = the angle MPR. Cor. 2. It is evident that the tangent at the vertex A is perpendicular to the axis. Tro?. VI. 8. The tangents at the extremities of a focal chord inter- sect at right angles in the directrix. Let PSQ be a focal chord) and let the tangent at Pmeet the directrix in Z. Join SZ ; then the angle ZSP is a right angle, (Prop. IV.) and .*. also the angle ZSQ is a right angle, .*. Z Q is the tangent at Q, (Prop. IV. Cor.) or the tangents at the extremities of the focal chord PSQ intersect in the directrix. Again, draw PM, QM' at right angles to the directrix ; then since MP, PZ — SP, PZ, each to each, and the angle MPZ = the angle SPZ, . \ the angle MZP = the angle SZP, . * . the angle SZP is half of the anjile SZM. So the angle SZQ is half of the angle SZM', .'. the angle PZ Q is half of the two SZAl and SZJl'. But the angles SZM and SZM' = two right angles, .'. the angle PZQ is a right angle, or the tangents at the extremities of a focal chord intersect at right angles in the directrix. 10 CONIC SECTIONS. Prop. VII. 9. If the tangent at any point P of a parabola meet the axis produced in the point T, and PN be the ordinate of the point P, then NT= 2 AN Join SP, and drawPilfat right angles to the directrix; then .'.the angle SPT = the angle MPT = the angle STP, .'. ST = SP But SP = PM = XN, .'. ST = XN But AS = AX, . • . the remainder AT = the remainder A N, .\NT = 2 AN. Def. The line NT is called the Suhtangent. 10. Dee. The line PG, drawn at right angles to PT, is called the Normal at the point P, and NG the Subnormal. Prop. VIII. If the normal at the point P of a parabola meet the axis in the point G } then NG = 2AS. CONIC SECTIONS. 11 Since the angle SPG = the complement of the angle SP P, and the angle SGP — the complement of the angle S TP, and also the angle SPT - the angle STP, {Prop. VII.) .-. the angle SPG = the angle SGP, .'. SG = SP. But SP = PM = XN 3 .\SG = XN. Taking away the common part SN 7 the remainder NG = SX — 2 AS. Prop. IX. 11. If PN be an ordinate to the parabola at the point P; then PiV 2 = 4AS.AN. Since TPG is a right angle, and PN perpendicular to TG ; . \ PN is a mean proportional between PVand NG ; or PN 2 = TN. NG. (Euclid, VI. 8 Cor.) But PiV = 2-4 JV, (Prop. VII.) and NG = 2AS, (Prop. VIII.) .\PN*=AAS.AN. Prop X. 12. If the tangent at any point P intersect the tangent at the vertex in Y, then S Y will bisect PT at right angles, and will be a mean proportional between SA and SP. Draw PVat right angles to the axis ; then since A Y is parallel to PN, .'. TY: YP:: TA : AN. But A T = ^ V, (Prop. VII.) .-. Tr=P7; and .-. SY, YP = SY, YT, each to each, and SP= ST } (Prop. VII.) .-. the angle SYP= the angle SYT. .'. #I r is perpendicular to PP. 12 CONIC SECTIONS. A^ain, since T YS is a right angle, and YA perpendicular to ST, .*. SY is a mean proportional between ST and SA ; or SY 2 = ST . SA, {Euclid, VI. 8 Cfcr.) But /ST = SP, (Prop. VII.) .-. £F 2 = £P. £A Cop.. If PJ/be drawn at right angles to the directrix, and M Y be joined, then since SP, PY = MP, PY, each to each, and the angle SPY= the angle MPY, (Prop. V.) .-.the angle SYP= the angle MYP, . \ /S Y y and Fif are in the same straight line. Prop. XI. 13. To draw a pair of tangents to a parabola from an external point. CONIC SECTIONS. 13 Let be the given external point. Join OS, and with centre and radius OS describe a circle, cutting the directrix in M and M', which it will always do, on whichever side of the directrix is situated, since is nearer to the directrix than to the focus. (Prop. II.) Draw MQ and M'Q' parallel to the axis meeting the parabola in Q and Q'. Join OQ, OQ' ; these will be the tangents required. Join SQ and SQ' ; then .-. OQ, QS = OQ, QM, each to each, and OS = OM 3 .'. the angle OQS = the angle OQM, ,\ OQ is the tangent at Q. (Prop. V.) So (?$' is the tangent at Q\ Prop. XII. 14. If from a point a pair of tangents OQ and OQ' be drawn to a parabola, the triangles OSQ, OSQ' will be similar, and (9$ will be a mean proportional between SQ and SQ'. Join SM, cutting OQ at right angles (Prop. X. Cor.) in the point Y; then since the angle SQO = the angle MQO, (Prop. X.) and the angle J/(?0 = the angle SMM', each of these angles being the complement of the angle QMY, .-. the angle SQO = the angle SMM". But the angle SMM' at the circumference is half the angle SOM' at the centre, and is therefore equal to the angle SOQ'. .*. the angle SQO = the angle SOQ'. So the angle £6>£ = the angle SQ'O, .-. the remaining angle OSQ = the remaining angle OSQ'. 14 CONIC SECTIONS. And therefore the triangle OSQ is similar to the triangle OSQ' .-. SQ : SO :: £0 : SQ', .-. ^C • #S'P . P V, (Prop. XV.) r.RQ.RX=4:SP .PK But RQ = PV. .\RX=4SP. Now when the circle becomes the circle of curvature at P, the points R and Q move up to and coincide with P, and the lines RX and PW become equal. Hence the chord of the circle of curvature parallel to the axis = 4#P. Cor. 1. If PU be the diameter of the circle of curvature, and PP'the chord through the focus; then o since the angle FPU = the angle 1VPU, (Prop. VIII.) .-.PF=PW=4:SP. Cor. 2. If #Fbe drawn at right angles to PT ; then the triangle P FIT is similar to SYP, .'.PU-.PF:: SP: >ST, otPU:4:SP::SP:SY. Prop. XX. If QVQ' be any ordinate to the diameter PV, the circle described through the three points P, Q, Q' will intersect the parabola in a fourth point, which depends only upon the position of P. Draw the ordinate PJSf, and produce it to meet the parabola in P' ; then, since the subtangent = 2 . AN. (Prop. VII.) the tangents at P and P' will meet the axis in the same point T. Draw PR parallel to TP', meeting the parabola in R, and QQ' inO; then PO.OBiQO.OQ' ::SP': SP. (Prop. XVII.) 24 CONIC SECTIONS. But SP=ST = SP', (Prop. VII.) .\ PO . OR = QO . OQ'. HeDce by the converse of Euclid III. Prop. 35, the point R is on the circle which passes through P, Q t Q '. Cor. 1. Since TP and TP' are equally inclined to the axis, the lines QQ\ PR, which are parallel respectively to TPand T P' , are also equally inclined to the axis. Cor. 2. When the point V is brought indefinitely near to P, QQ' coincides with the tangent to the parabola at P, and becomes also a tangent to the circle at P, since Q and Q' are indefinitely near to each other. The circle therefore becomes the circle of curvature at the point P. Hence if PR be drawn parallel to the tangent at P', or be equally inclined to the axis with PT, it will meet the parabola in the point where the circle of curvature at P intersects the parabola. PEOBLEMS ON THE PARABOLA. 1. The diameter of the circle described about the triangle BA G is equal to 5 AS. (See fig. Prop, III.) 2. If from the point G, G K be drawn at right angles to SP, then PK = 2 A 8. (See fig. Prop. VII.) 3. If the triangle SPG is equilateral, then SP is equal to the latus rectum. (See fig. Prop. VII.) 4. PQ is a common tangent to a parabola and the circle described on the latus rectum as diameter ; prove that SP and SQ make equal angles with the latus rectum. 5. Prove that PY . PZ = SF% and that PY . YZ= AS. SP. (See fig. Prop. VII.) 6. If PL be drawn at right angles to A P. meeting the axis in L, and PiV be the ordinate of P, then ^\ r P = 4 ,.4$. 7. The tangent at any point P of a parabola meets the directrix and latus rectum produced in points equally distant from the focus. 8. Prove that NY = TY, and that TP . TY = TS. TN. (See fig. Prop. Mil.) 9. If a circle be described about the triangle SPN, the tangent to it from A = I PN. (See fig. Prop. VII.) 10. If the ordinate of a point P bisect the subnormal of P', the ordinate of P is equal to the normal of P' . 11. If from any point on the tangent to a parabola a line be drawn touching the parabola, the angle between this line and the line to the focus from the same point is constant. 26 CONIC SECTIONS. 12. A circle and parabola have the same vertex and axis. BA ' G is the double ordinate of the parabola which touches the circle at A' , the extremity of the diameter through the vertex A. PP' is any other ordinate of the parabola parallel to this, meeting the axis in JV, and A B produced in R ; prove that the rectangle RP . RP is proportional to the square of the tangent drawn from N to the circle. 13. Draw a parabola to touch a given circle at a given point, and such that its axis may touch the same circle in another given point. 14. If from the point of contact of a tangent to a parabola a chord be drawn, and another line be drawn parallel to the axis meeting the chord, tangent, and curve, this line will be divided by them in the same ratio as it divides the chord. 15. If the diameter PFmeet the directrix in 0, and the chord drawn through the focus parallel to the tangent at P in V, prove that VP = PO. 16. Prove that the locus of the intersection of a diameter PFwith the chord drawn through the focus parallel to the tangent at P is a parabola. 17. If a circle and parabola have a common tangent at P, and intersect in Q and R ; and Q V, UR be drawn parallel to the axis of the parabola meeting the circle in V and U respectively, then V U is parallel to the tangent at P. 18. AB and A G are two lines at right angles to each other. From a fixed point G on A G, GR is drawn parallel to AB. OnAR, produced if necessary, P is taken such that the perpendicular PN upon A B is equal to GR. Prove that the curve traced out by P is a parabola. 19. If from a point P of a circle PG be drawn to the centre; and R be the middle point of the chord PQ drawn parallel to a fixed diameter A GB, then the curve traced out by the intersection of CP and A R is a parabola. 20. If two equal tangents Q, Q\ be cut by a third tangent, their alternate segments are equal. CONIC SECTIONS. 27 21. E is the centre of the circle described about the triangle O OQQ'. Prove that the circle described about the triangle QEQ' will pass through the focus. [Sec fig. Prop. XIII.) 22. PSp is any focal chord of a parabola. Prove that AP, Ap will meet the latus rectum in two points Q, q, whose distances from the focus are equal to the ordinates of p and P. 23. PSp is a focal chord of a parabola, RDr the directrix meeting the axis in D ; and Q any point on the curve. Prove that if QP, Qp be produced to meet the directrix in R, r, half the latus rectum is a mean proportional between PR, Br. 24. OP and Q are two tangents to a parabola. On Q produced, Q' is taken equal to Q ; prove that OS . PQ = OP. OQ. 25. If QD be drawn at right angles to the diameter PV, then QD* = 4:AS.&V. 26. If through any point on the axis of a parabola a chord PO Q be drawn, and PM, QN be the ordinates of the points Pand Q, prove that AM. AN = A 2 . 27. If AP and A Q be drawn at right angles to each other from the vertex of a parabola, and PM, QN be the ordinates of P and Q, prove that the latus rectum is a mean proportional between A M and AN. 28. OA Pis the sector of a circle whose centre is 0. If the radius OA remain fixed while the angle A OP changes the centre of the circle inscribed in the sector, AOP will trace out a parabola. 29. QSQ f is a focal chord parallel to ^1P; PN, QM, Q'M are the ordinates of P, Q, and Q'. Prove that SM 2 = AM . ^iVand that MM' = AP. 30. PQ, PQ' are drawn from any point P cutting the ordinates Q' V, QVinR' and R, prove that VR is to V R' in the triplicate ratio of Q V to Q' V. 28 CONIC SECTIONS. 31. On a chord of a parabola as diameter a circle is described cutting the parabola again in two points. If these points be joined, the portion of the axis between the two chords is equal to the latus rectum. 32. If OQ, Q' be a pair of tangents to a parabola, and the chord Q Q' be a normal to the curve at Q, then OQ is bisected by the directrix. 33. Two equal parabolas having the same focus and their axes in contrary directions intersect at right angles. 34. The radius of curvature at the extremity of the latus rectum is equal to twice the normal. 35. If from any point P of a parabola PF and PH be drawn making equal angles with the normal PG, then SG 2 —- SF . SH. 36. If a triangle be inscribed in a parabola, the points when the sides produced meet the tangents at the opposite angles are in the same straight line. 37. If the tangents Q, Q' be cut by a third tangent in R, R', prove that OR:RQ:: R' Q' : OR'. 38. If from the vertex of a parabola chords be drawn at right angles to one another, and on them a rectangle be described, the curve traced out by the further angle is a parabola. 39. Prove that 2PY is a mean proportional between AP and the chord of the circle of curvature at the point P of the parabola drawn through the vertex A. {See fig. Prop. VII.) 40. If a circle described upon the chord of a parabola as diameter meet the directrix it also touches it, and all chords for which this is possible intersect in a point. 41. If a parabola roll upon another equal parabola, the vertices originally coinciding, the focus traces out the directrix. 42. The circle of curvature at the extremity of the latii6 rectum intersects the parabola on the diameter of curvature passing through the point of contact. CHAPTER II. THE ELLIPSE. 22. Def. The Ellipse is the curve traced out by a point which moves in such a manner that its distance from a given fixed point continually bears the same ratio, less than unity, to its distance from a given fixed line. (See Introdiietion) PliOP. I. The focus and directrix of an ellipse being given, to find any number of points on the curve. Let S be the focus, and MX the directrix. Draw SX at right angles to the directrix, and divide SX in the point A, so that SA may be to AX in the given fixed ratio less than unity ; then A is a point on the curve. On XS produced take a point A ', such that SA' : A'X :: SA : AX; then A' will also be a point on the curve. On the directrix take any point M ; and through 31 and S draw the line MYSY' meeting AY and A'Y\ drawn at right angles to A A ' in the points Y and Y\ On YY' as diameter describe a circle, and draw 3IPP' parallel to AA\ cutting the circle in the points P and P' ; P and P' will be points on the ellipse. 30 CONIC SECTIOKS. Join FY, PY\ SP; then since SY : YM :: SA : AX, (Euclid, VI. 2.) and SY' : YM :: SA' : A'X, (Euclid, VI. 2.) .-. SY : FJf :: £F' : F'if; or, alternately, #F : SY' :: FJJf : F'if, and the angle YP Y in a semicircle is a right angle, .-. PY bisects the angle SPM* .-. &P : PM :: £F : FJ£ :: SA : ^tX So we may show that SP : P'M :: £F : YM, :: &^ : ^X, .. P and P' are points on the curve. * For, if not, make the angle YPs equal to YPM ; then sY : YM :: sP : PAf. (.EttcZta, VI. 3.) And since, if P F bisect sPM, P Y', being at right angles to P F, also bisects the angle sPM', .\ sY' : Y'M :: sP : PM. {Euclid, VI. A.) Hence sY : FAT :: »F : 3 r 'Af, or « Y : * Z' :: YM : V'Af, . ". the points S and s coincide. CONIC SECTIONS. 31 In the same way, by taking other points on the directrix, we may obtain as many more points on the curve as we please. Cor. 1. Since, corresponding to every point P on the curve, there is a point P' situated in precisely the same manner with respect to A ' Y' as P is with respect to A Y, it is clear that if we make A 'S' equal to A S, and A ' X' equal to A X, and draw X'M' at right angles to AX', the curve could be equally well described with S' as focus and MX' as directrix. The ellipse is therefore symmetrical, not only with respect to the line A A', but also with respect to the line OC drawn through the middle point of YY' at right angles to and bisecting A A '. Cor. 2. The line OP will bisect the angle SPS'. Let OP meet SS' in G. Produce MP to meet X'M' in M', and draw OM' passing through the focus S' ; then SP : PM :: S'P PM', or, alternately, SP : S'P :: PM : PM'. Again, SG : PM:: S'G : PM', or, alternately, SG : S'G :: PM : PM\ • . from (1) anc 1(2) SP : S'P :: SG I O Cr, (1) (2) .\ PG bisects the angle SPS'. (Euclid, VI. 3.) It will be shown hereafter (Prop. XI.) that the normal to the ellipse at the point P also bisects the angle SPS'. Hence the ellipse and circle have the same tangent at the point P. The ellipse will consequently touch all the infinite series of circles which can be described in the same manner as the one in the figure by taking different points on the directrix. 32 CONIC SECTIONS. Prop. II. 23. If Che the middle point of AA', then CA is a mean proportional between CS and CX, or CS . CX = CA 2 . {See fig. Prop. III.) Since SA' : AX :: SA : AX. Alternately SA' : SA :: ^'X : AX, .-. £J' + SA : &4 :: .4X + AX : ^X; or .4.4' : SA :: XX : AX, .-. ^^T : XX :: SA : A X } or CA : CX :: £4 : AX. (I)* Again, £^4' : SA :: ^'X : AX, .'. SA' - SA : SA :: J/X - ^1X : ^1X; or ££' : SA :: ^' : AX. Alternately SS' : .4.4' :: SA : .4X: or C£ : G'J. :: SA : JZ (2) Hence from (1) and (2) CA : CX :: CS : CA, .-. C.4 2 = CX . 6'£; or 6'^4 is a mean proportional between CS and CX. Cor. Since the three lines CS, CA, CX are proportional, therefore, by the definition of duplicate ratio and Euclid, VI. 20 Cor. CS : CX : CS 2 : CA 2 . (3) Prop. III. 24. If P be any point on the ellipse, then SP+ S'P= A A. * N.B. The results (1), (2), (3) should be remembered, as they will frequently be referred to. CONIC SECTIONS. : XX', : XX', A A' : XX'. Since £P : PM :: /SM : ^X, and £J : ^1X :: ^' : IP, (Prop. II.) .-. SP: PM:: AA' : So 8'P:PM'::AA' : .-. £P + S'P'.FM+PM' . But pj/ 4 pjt = jot = xx\ .-. SP + S'P= AA'. Cor. 1. By means of this property the ellipse maybe prac- tically described and the form of the curve determined. Let a string, equal in length to A A', have its ends fastened to two points S and 8' ; and let it be kept stretched by means of the point of a pencil at P; then since SP 4 S P will be always equal to A A', the point P will trace out the ellipse. COR. 2. The line A A! is the longest line that can be drawn in the ellipse. For, if any other line PQ be drawn, then SP + SQ> PQ, and S'P+ S'Q> PQ, .-.SP+ S'P+ SQ + S'Q>2PQ, ot AA' > PQ. 25. Def. If BCB' be drawn at right angles to AC A', meeting the ellipse in B and B' , it will be seen further on {Prop. XIII. Cor. 2) that BCB' is the shortest chord that can be drawn through the centre of the ellipse. (Sec fig. Prop. IV.) 34 CONIC SECTIONS. A A' is called the Major Axis and BB' the Minor Axis of the ellipse. In most geometrical treatises the ellipse is defined as the curve traced out by a point which moves in such a manner that the sum of its distances from two fixed points is always the same ; but it appears that the properties of the curve are more clearly exhibited by defining it in a manner analogous to the parabola, and deducing immediately from that definition the property in question. Having now shown that one definition necessarily includes the other, we are at liberty in our future investigations to make use of whichever property is most convenient. Pkop. IV. 26. If B be the semi-minor axis of the ellipse, then BC 2 = CA 2 - CS 2 ; and if SL be the semi-latus rectum, SL .AC = BC\ Join SB, S'B; then since SB + S'B and that SB .'. SB But B C 2 r.BC 2 A A', S'B, AC. SB 2 CA 2 (Prop. III.) CS 2 , cs\ CONIC SECTIONS. 35 Again, SL : SX : : SA : A X, :: CS: CA, (Prop. II.) .'. SL.AC= CS. SX, = -^. ii.) = BC 2 . Prop. V. 27. The sum of the distances of any point from the foci of an ellipse will be less or greater than A A' according as the point is inside or outside the ellipse. a s (1) Let Q be a point inside the ellipse. Join S Q, S'Q\ and produce S Q to meet the ellipse in P ; join S' P\ then since S'P+ QP> S' Q, .'. S'P+ SP > S'Q + SQ. But S'P + SP= A A', [Prop. III.) .'. SQ+ SQ' S'P, .-. S'Q+ SQ> SP+S'P, But SP + S'P= AA\ (Prop. III.) .-. SQ+ S'Q> AA'. D 2 36 CONIC SECTIONS. Cor. Conversely, a point will be inside or outside the ellipse, according as the sum of its distances from the foci is less or greater than AA. 28. Def. If a point P' be taken on the ellipse near to P, (see fig. Prop. VI.) and PP' be joined, the line PP' produced, in the limiting position which it assumes when P' is made to approach indefinitely near to P, is called the Tangent to the ellipse at the point P. Prop. VI. If the tangent to the ellipse at any point P intersect the directrix in the point Z, and if 8 be the focus corresponding to the directrix on which Z is situated, then SZ will be at right angles to SP. Let P' be a point on the ellipse near to P. Draw the chord PP\ and produce it to meet the directrix in Z; join SZ. Draw PM, P'M' at right angles at the directrix, and join SP, SP'. CONIC SECTIONS. 37 Produce PS to meet the ellipse in the point Q ; then since the triangles ZMP, ZM'P' are similar, .-. ZP : ZP' :: MP : M'P', :: SP : SP', .-. £^bisects the angle P'SQ. {Euclid, VI. Prop. A.) Now when P' is indefinitely near to P, and PP' becomes the tangent at the point P, the angle PSP' becomes inde- finitely small, while the angle QSP' approaches two right angles: and therefore the angle ZSP' being half of the angle P' SQ, becomes ultimately a right angle. Hence when PZ becomes the tangent at the point P, the angle ZSP is a right angle, or SZ is perpendicular to SP. Cor. 1. Conversely, if SZ be drawn at right angles to SP, meeting the directrix in Z, and PZ be joined, PZ will be the tangent at P. Cor. 2. If ZP be produced to meet the other directrix on the point Z' and S'Z' be joined, then S'Z' is at right angles to S' P. Cor. 3. The tangents at the extremities of the latus rectum or double ordinate through the focus meet the axis produced in the point X. Prop. VII. The tangent to the ellipse at any point P makes equal angles with the focal distances &Pand S' P. Let the tangent at P meet the directrices in Z and Z '. Draw MPM' at right angles to the directrices, meeting them in ilf and M' respectively ; join SZ, S'Z' ; then SP : PM :: S'P : PM' ; and since the triangles MPZ, M'PZ ', are similar, PM : PZ : : PM' : PZ\ .-. SP : PZ :: S'P : PZ'. {Ex ceqiiali.) 33 CONIC SECTIONS. Now in the triangles SPZ, S' PZ', because the sides about, the angles SPZ, S' PZ' are proportional, and the angles PSZ, PS' Z' are equal, being right angles, and the angles SZP, S'Z'P&ve each less than a right angle, .*. the triangles SPZ and S' ' PZ' are similar, {Euclid, VI. 7) .-. the angle SPZ = the angle S'PZ'. Cor. If S'P be produced to W; then the angle SPZ = the angle WPZ. Prop. VIII. The tangents at the extremities of a focal chord intersect in the directrix. Let PS Q Toe a focal chord, and let the tangent at Pmeet the directrix in Z. Join SZ ; then CONIC SECTIONS. 39 the angle Z SP is a right angle, (Prop. VI.) and .*. also the angle ZSQ is a right angle, .". ZQ is the tangent at Q ; {Prop. VI. Cor. 1) or the tangents at the extremities of a focal chord intersect in the directrix. Prop. IX. 29. If the tangent at P meet the axis major produced in T, and PX be the ordinate of the point P, then CT . CN = GA\ Draw MPM' parallel to the axis major meeting the direc- trices in M and M' ; and produce S'Pto W\ then, since PT bisects the angle SPW, {Prop. VII. Cor.) .-. S'T : ST :: S'P : SP, {Euclid, VI. A.) PM ' : PM, X'N : X2V, . &T+ ST : ST - ST :: X'.V + XN : Z'JV - ZiV; or 2CT : 2 CS :: 2 <7X : 2 CJV, or CT : CS :: CX : <7iV, ... or . on = cs . ex, = CA\ (Prop. II.) 40 CONIC SECTIONS. Prop. X. If on the major axis of an ellipse as diameter a circle be described and a common ordinate NPQ be drawn meeting the ellipse in P and the circle in Q, then the tangents to the ellipse and circle respectively at the points P and Q will meet the major axis produced in the same point. Let the tangent to the ellipse at P meet the major axis produced in T; join C Q, QT; then, by the last Proposition, CT . CN = CA 2 = CQ\ . • . the angle C Q T is a right angle. And therefore Q T is the tangent to the circle at Q ; or the tangents at P and Q meet the major axis produced in the same point T. The circle described on A A' as diameter is called the Auxiliary Circle on account of the important aid that it affords in investigating the properties of the ellipse. 30. Def. The line PG, drawn at right angles to the tangent P T, is called the Normal to the ellipse at the point P. Prop. XL If the normal at P meet the major axis in the point G then SG : SP :: CS : CA, CONIC SECTIONS. 41 Since PG is at right angles to TPt, .-. the angle GPT= the angle GPt. But the angle SPT= the angle S'Pt, (Prop. VII.) .-. the angle SPG = the angle S'PG, or PG bisects the angle SPS', .'.SG : S'G :: SP : £'P, (JStaftZ, VI. 3.) SG : £G + £'G or #G : ££' or SG : SP or ££ : SP SP : SP + S'P SP : AA' ; .SS' : AA') CS : CA. Con. Hence also, S'G : S'P :: CS : CA. Prop. XII 31. If the normal at P meet the major axis in G, and PX be the ordinate at the point P, then (see fig. Prop. XI.) NG : XC :: BC 1 : AC 2 . Draw MPM' parallel to the axis meeting the directrices in J^and M' ; join SP, S'P', then, since PG bisects the ano-le SPS', (Prop. XL) .*. S'G : SG .-. S'G-SG: S'G + SG or 2 CG : SS' S'P : SP, PM' : PM, XX : XX, X'F-XN : X'N+ XX; 2 CN : XX'. 42 CONIC SECTIONS. Alternately, 2 CO or CG .'. CN— CG : ovNG : 2CN :: SS' : ZJT; (7iV :: (75 : CJT, :: CS 2 : C^t 2 , CiV :: CA 2 - CS 2 : : CN :: PC 2 : ^C 2 . Prop. XIII. {Prop. II. Cor.) C.4 2 ; 32. If Pi\ r be the ordinate of any point P on the ellipse ; then PN 2 : AN . -4'iV:: BC 2 : ^C 2 . N G C Produce NP to meet the auxiliary circle in the point Q, and draw the tangents FT, QT meeting the major axis pro- duced in the point T. {Prop. X.) Join CQ, and let the normal at F meet the ellipse in G then, by the last Proposition, NG : CN :: BC" : AC 2 . And rectangles of the same altitude are to another as their bases, .-. TN . NG : TN . CN :: BC 2 : AC 2 ; or PN 2 : QN 2 :: BC 2 : AC 2 . [Euclid, VI. 8, Cbr.) But QN 2 = AN. A'N, since the angle AQA' in a semicircle is a right angle, .-. PiV^ : AN . A'N :: BC 2 : .4 C 2 . Cor. 1. Also PiV : §# :: BC : A (7. This result is the basis of many of the future Propositions of the ellipse. CONIC SECTIONS. 43 Cor. 2. Since PN 2 : QN 2 :: BC 2 : AC 2 , .\ PiV 2 : ^C 2 - CiP :: B C 2 : ^ C 2 , .-. FN 2 : AC 2 - CN 2 - PN 2 :: BC 2 : AC 2 - BC 2 , or PN 2 : AC 2 - CP 2 :: BC 2 : 4 <7 2 - BC 2 . Now PiV 2 is always less than B C 2 , .'. CP 2 is greater than BC 2 , .'. BC is the shortest line that can be drawn to the ellipse from the centre. Prop. XIV. If the tangent at any point P of an ellipse meet the minor axis CB produced in t, and Pn be drawn at right angles to CB; then Ct . On = B C\ Draw the common ordinate NPQ to the ellipse and the auxiliary circle ; and let the tangents at P and Q to the ellipse and circle respectively meet the major axis produced in T {Prop. X.) and the minor axis produced in t and TJ. Join CQ meeting Pn in P; then since PR is parallel to CX, CE : CQ :: PIST : QN. :: BC : AC. (Prop. XIII. Cor. 1.) 44 CONIC SECTIONS. But CQ = AC, .'. CR = BC. (1) Again, joining Rt t Ct : CU :: PN : QN } :: CM : CQ, . •. Et is parallel to ^ . *. CRt is a right angle, .\Ct . Cn= CE 2 {Euclid, VI. 8, Cor.) But CR = B C, (1) .-. Ct . Cn = BC 2 . This proposition also admits of a demonstration similar to that given for the corresponding property of the hyperbola. Prop. XV. 33. If from the foci S and S', SYand S' Y' are drawn at right angles to the tangent at P, then Y and Y' are on the circumference of the auxiliary circle, and SY . S'Y' = BC\ CONIC SECTIONS. 45 Join SP, S' P, and produce SY and S'P to meet in IF; and join CF; then since the angle SPY = the angle WPY, (Prop. VII. CV.) and the angle £FP = the angle WYP, and the side PY is common to the triangles SPY, WPY, .*. the triangle SPY = the triangle TFPFin all respects, . •. SP = P W, .-. SP + S'P= S'W. But SP + S'P = A A', (Prop. III.) .-. S'W= AA\ Again v £C = £'C and SY= YW, .-. SC : S'C :: SY : FTF, .\ CY is parallel to S' W, .'. CY : £'TF :: CS : ££', .-. CY = \S'W= CA. So CF' = CA, .-. Fand Y' are points on the auxiliary circle. Next let Y S be produced to meet the auxiliary circle in Z, and join Z Y' ; then since the angle ZYY' is a right angle, . • . Z Y' passes through the centre C, .-. the angle SCZ= the angle S r CY'. .' . SZ = o Y , .-. £F . S'Y' = SY . &£ = A&.A'8, (Euclid, III. 35) = 6L4 2 - C£ 2 , (JWwf, II. 5) = PC 2 . (Prop. IV.) Con. If OP be drawn parallel to the tangent at P, meeting S'P in P ; then 46 CONIC SECTIONS. since the figure CYPE is a parallelogram, .-. PE = CY = AC. Pkop. XVI. 34. To draw a pair of tangents to an ellipse from an external point 0. With centre S' and radius equal to A A' descnoe a circle. Join OS, OS' ; and let SO or S'O produced meet the circle in the point I, Now, if be a point outside the circle MIM\ it is evident that OS is greater than 01 ; and if be inside the circle, since OS + OS' > A A' or S'l (Prop. V.) .*. OS> 01. With centre and radius OS describe another circle cutting the former in the points M and M\ which it will always do, since OS is greater than OL CONIC SECTIONS. 47 Join S'M, S'M', meeting the ellipse in the points P and P'. Join OP, OP' ; these will be the tangents required. Join SP, SP' ; then, since SP + S'P = AA' = S'M, ;•. sp = pm. And v SP, PO = MP, PO, eacli to each, and OS = 0M } ,\ the angle OPS = the angle OPM, .'. OP is the tangent at P. (Prop. VII. Cor.) So OP' is the tangent at P' . Prop. XVII. If from a point a pair of tangents OP, OP' be drawn to an ellipse, then OP and OP' will subtend equal angles at either focus. Join SP, S'P; SP, S'P' ; and produce S'P, S'P' to M and if, making PM equal to /S'P, and FM' equal to #P. JoinOiH, QM'\ OS, OS'. Then since OP, PS = OP, PM, each to each. and the angle OPS = the angle OPM, {Prop. VII. Cor.) .-. 0£= 0ilf, and the angle 0£P= the angle OMP. So <9£ = OM', and the angle 0£P' = the angle OM'P', .-. 0M~ OM'. Again, v S'Jf = S'P + SP= A A', and S'M' = S'P' + £P' = ^', .•.S'M-S'M', And v 05', tf'Jf = 0>S", S'M', each to each, and OM = 0JT, .-. the angle OS'M = the angle 0£'Jtf"', and the angle 0Jlf£' = the angle 0JT£'. 48 CONIC SECTIONS. But the angle OMS' = the angle OSP, and the angle OM'S' = the angle OSP', .'. the angle OSP = the angle OSP', .'. OP and OP' subtend equal angles at either focus. Prop. XVIII. 35. If from an external point a pair of tangents OQ, OQ' be drawn to an ellipse, and CO be joined meeting the chord QQ' in V, and the ellipse in P ; then (1.) QQ' will be bisected in V. (2.) The tangent at P will be parallel to QQ'. (3.) CP will be a mean proportional between CV said CO. Produce OQ, OQ' to meet the major axis produced in T and T'. Draw the ordinates NQ, N'Q' , and produce them to meet the circle in q and q. CONIC SECTIONS. 49 Then Tq and T'q will be tangents to the auxiliary circle. [Prop. X.) Let Tq and T'q be produced to meet in o ; join Co meeting the chord qq in v, and the circle in|?. Now, since the corresponding ordinates of the ellipse and auxiliary circle are in the constant ratio of BC to A C, the three lines ol,pm, vn drawn at right angles to AA will pass through the points 0, P, V respectively. For, according as is the point where ol meets TQ or T Q' we shall have 10 : lo :: NQ : Nq, :: BC : AC; or 10 : lo :: N' Q' : N'q', :: BC : yiC, .-. Oo is perpendicular to A A. So Pp and Fv are perpendicular to A A ' , /. Oo, Pp, Vv are parallel. Hence (1.) QV : VQ' :: qv : ^'. But g"y = t'#/ from the circle, .-. QV= VQ'; or QQ' is bisected in V. (2.) Since iV£ : Nq :: #', .'. CV : CP :: CP : CO, .-. (7(9 . (7F= CP 2 . Cor. From this it is manifest that if any number of chords be drawn parallel to each other in an ellipse, their middle points will all lie on the line drawn from the centre to the point where the tangent parallel to the chord meets the ellipse. Def. The line PCP' drawn through the centre of an ellipse and meeting the curve in P and P' , is called a Diameter. The diameter consequently bisects all chords parallel to the tangents at its extremities ; and the tangents at the extremities of any chord will intersect the diameter corre- sponding to that chord in the same point. 36. Def. If CD he drawn parallel to the tangent at P } then CD is said to be conjugate to CP. Pjrop. XIX. In the ellipse if CD be cod jugate to CP, then will CP be conjugate to CD. Draw the ordinates PN, DP, and produce them to meet the auxiliary circle in the points p, d. Join CP, Cp ; CD, Cd ; and draw the tangents TP, Tp ; T D, T'd. Now, since CD is parallel to PT, CONIC SECTIONS. 51 .*. the triangle PNT is similar to the triangle DR C. .'. TN : CR :: PN : DR, .-. Np : Rd, (Prop. XIII. Cor.) .*. Tp is parallel to 6VZ, .*. the angle^CVZ is a right angle, .*. Cp is parallel to T'd, .•. the triangle p C N is similar to the triangle dT f R, .-. NC : RT :: Np : J?£ :: iVT: RD, .'. CP is parallel to D T', .'. CPis conjugate to CD. Cor. Since CRd and ONp are each similar to dR T , (Euclid, VI. 8j .\ the triangle CRd is similar to the triangle CNp, and the side Cd = the side Cp, .'. the triangle CRd — the triangle CJSfp in all respects, .-. (7iV - ita, and CR = Np. Hence DR : OA r :: DP : ita, :: PC : -4(7; also PN : CR :: PN : Np, :: PC : AC. Peop. XX. 37. If C'P and 6' 7) he conjugate semi-diameters, and PN DR he the ordinates of the points P and D ; then ' (1.) CiP + CR 2 = AC\ I (2.) PJT + Z>i2 2 = P<7< f (3.) CP 2 + CD' 2 = AC 2 + DC 2 . E 2 52 CONIC SECTIONS. BC : AC, BC 2 : AC 2 . BC* : AC*, Produce JVP, RD to meet the auxiliary circle in the points p, d; then CJST = Ed, (Prop. XIX. Cor.) .-. CN 2 + CR 2 = Ed 2 + £P 2 , = Cd 2 , = CU 2 . Again, PN : JVjp .-. pa^ 2 : jsy So Z>i^ 2 : Ed 2 .-. PiST* + Pi^ a : iVp* + Pd 2 :: P<7 : ^C 2 ; butJV>*+ P^ a = CP 2 + CN 2 , = AC 2 , .-. PN 2 + PP 2 = PC 2 , and CiT + OP 2 = J C 2 , .-. OP* + CD 2 = AC 2 + PG 72 . 38. Def. Aline §F drawn parallel to the tangent at P, and meeting CP in V, is called an Ordinate to the diameter CP. Pr.or. XXL If Q V be any ordinate to the diameter PCP', and CD be conjugate to CP; then C^ 2 : PV . P'V :: CP 2 : CP 2 . CONIC SECTIONS. 53 Draw the tangent UQW meeting CP and CD produced in U and IK; and draw QJR parallel to CP, meeting CD in R. jSTow, since CR : CD .'. CR' 2 : CD 2 or QV 2 : CD 2 CD : C W, {Prop. XVIII.) CR : CW t {Euclid, VI. 20 Cor.) UV : 6**7. Again, since CU : CP :: CP : or, {Prop. XVIII.) .-. CU : CF :: CP : CP, (J5Wt#, VI. 20 Cor.). ... CU- CV : CU :: CP 2 - - CF 2 : CP 2 , or UV : C27 :: PF . P'V : CP 2 . Hence Q I 7 ^ : CP 2 :: PV . P'F : CP\ or #F 2 = PV -P'V -- CD 2 : CP\ Prop. XXII. 39. The area of any parallelogram formed by drawing tangents to an ellipse at the extremities of a pair of conjugate 54 CONIC SECTIONS. diameters is equal to the rectangle contained by the axes of the ellipse. Let PCP', DCD' be a pair of conjugate diameters, and let a parallelogram be formed by drawing tangents at the points P, P', D, D'. Let the tangent at P meet C A produced in T\ join D' T. Draw the ordinates PN, DR, D'B'; then since PT is parallel to CD', the parallelogram PD' is double the triangle CTD', and therefore equal to the rectangle contained by C T and D'B'. Now V'R' : CN :: BC : A 0, (Prop. XIX. Cor.) .-. CT . D'B' : CT . CF :: BC : AC, (Euclid, VI. 1.) :: BC . AC : AC 2 . (Euclid, VI. 1.) But CT . CN = A C\ .-. CT . D'B' = AC . BC, c*. the parallelogram LL' = 4 the parallelogram PD', = 4AC. BC, = AA' . BB\ CONIC SECTIONS. 55 Cor. If PF bo drawn at right angles to D CD' meeting CD' in F; then PF . CD' = area of parallelogram PD', Prop. XXIII. 40. If CP and OZ> be conjugate diameters, and PF be drawn at right angles to CD meeting CA in 6 r , then A' Draw the ordinate PN, and produce it to meet CD' in A'. Also draw Pn at right angles to CB, and let the tangent at P meet 07? produced in t. Now since the angles at JVand .Fare right angles, it i^ evident that a circle may be described about the quadrilateral figure NKFG ; .-. PG . PF= PX. PK, (Euclid, III. 3G Cor.) = Ct . Cn, = BC 2 . {Prop. XIV.) 5(3 CONIC SECTIONS. Prop. XXIV. 41. If Pbe auy point on the ellipse, and CD be conjugate to CP, then SP. S'P = CD 2 . Draw the normal PG and produce it to meet CD' in F\ tli en since CD' is parallel to the tangent at P, .'. PPis at right angles to CD', r.PF. CD = AC.BC, (Prop. XXII. Cor.) and PF . PG = £ C 2 = BC.B C, (Prop XXIII.) .-. C7Z>: PGv.AC.BC. (1) Again, &P : SG:: CA: CS, (Prop. XL) £'P: £'G :: CA : C& (JVip. XL) Compounding SP . £'P : ££ . £'67 :: CU 2 : G T £ 2 , .-. £P. £'P: SP. S'P- SG. S'G:: CA 2 : CA 2 - CS 2 . But £P. S'P- SG . S'G = PG 2 , {Euclid, VI. Prop. 13) .-. #P. S'P:PG 2 :: C./1 2 : PC 2 . But from (1) CD 2 : PG 2 :: CA 2 :BC .-. .S'P. £'P = CD 2 . This proposition may also be very easily deduced from Prop. XV. CONIC SECTIONS. 57 Peop. XXV. 42. The area of the ellipse is to the area of the auxiliary circle as B C to A C. N N> C Let PN and P'N' be two ordinates of the ellipse near together. Produce NP, N'P\ to meet the auxiliary circle in Q and Q '. Draw Pm, Qn, perpendicular to Q'N'. Then the parallelogram PN' : tlie parallelogram QN* :: PN: QN, ::BC: AC. And the same will be true for all the parallelograms that can be similarly described in the ellipse and auxiliary circle. Hence the sum of all the parallelograms inscribed in the ellipse is to the sum of all the parallelograms inscribed in the circle as B C to A C. And this holds however the number of parallelograms be increased. But when the number of parallelograms is increased, and the breadth of each diminished indefinitely, the sum of the parallelograms inscribed in the ellipse will be equal to the area of the ellipse, and the sum of those inscribed in the circle to the area of the circle. Hence the area of the ellipse : the area of the circle : : BC : AG. dS CONIC SECTIONS. 43. Def. If with, a point on the normal at P as centre, and OP as radius, a circle be described touching the ellipse at P, and cutting it in Q ; then when the point Q is made to approach indefinitely near to P, the circle is called the Circle of Curvature at the point P. Piiop. XXVI. If PII be the chord of the circle of curvature at the point P of an ellipse, which passes through the centre ; then PII. CP=2 CD\ Let PT be the tangent, and PG the normal at the point P. T With centre 0, and radius OP, describe a circle cutting the ellipse in the point Q. Draw RQW parallel to CP, meeting the circle in W, and TP produced in R. CONIC SECTIONS. 59 Also draw Q V parallel to PR, meeting the diameter PP' in V; then since IIP touches the circle at P, .-. E Q . B W = PR 2 , {Euclid, III. 36) ovPV. RW = QV\ But Q V 2 : P V. P' V :: CD 2 : OP 2 , (Prop. XXL) .-. PV.RtViPV. P'V:: CD 2 : CP 2 , or RW: P'V:: CD 2 : CP 2 . ISTow, when the circle becomes the circle of curvature at P the points R and Q move up to, and coincide with P, and the lines R W and PH become equal, while P'V becomes equal to PP' or 2 CP. Hence, PH:2CP :: CD 2 : CP 2 , :. PH. CP : 2 CP' :: 2 CD 1 : 2CP\ ..PH. CP= 2 CI) 2 . Pkop. XXVII. If PTJho, the diameter of the circle of curvature at the point P of the ellipse, and PF be drawn at right angles to CD\ then PU.PF= 2CD\ Since the triangle PHU is similar to the triangle PFC, ..PU.PH:: CP-.PF, ..PU. PF=PH. CP, = 2CD 2 . {Prop. XXVI.) Pkop. XXVIII. If PI be the chord of the circle of curvature through the focus of the ellipse ; then PI. AC= 2CD\ 60 CONIC SECTIONS. Let PI meet CD in E \ then, since the triangles PIU and PEF axe similar, .-. PI : PIT :: PF : PP. But PE = A C, (Prop. XV Cor.) .'. PI : PIT :: PF : AC, .-. PI . AC= PU . Pi 7 , = 2 OZ> 2 . (Prop. XXVII.) Prop. XXIX. 44. If two chords of an ellipse intersect one another, the rectangles contained by their segments are proportional to the squares of the diameters parallel to them. Let POP' he any chord drawn through the point 0, and let CD he the semi-diameter parallel to it. Draw the ordinates NP, N'P', MD, and produce them to meet the auxiliary circle in Q, Q\ D' ; then since NP : NQ :: N'P' : N'Q', (Prop. XIII. Cor.) it is evident that PP' and QQ' will meet the axis produced in the same point T. MA' Also since NP : NQ :: MD : MD', (Prop. XIII. Cor.) and TPP' is parallel to CD, .-. TQQ' is parallel to CD'. CONIC SECTIONS. CI Draw EO parallel QQ' in 0' \ then to NQ or JSTQ' , and produce it to meet DO : QO' :: TO : TO', and P'O ••Q'O' :: TO : TO', .'.DO . P'O : QO' . Q'O':: TO* CD 2 CD'' : T0'\ : CD' 2 , : AC 2 . Alternately, DO . D'O : CD 2 :: QO' . Q'O' : AC 2 . Again, if through the point any other chord pOp be drawn, since EO : EO' :: EC : ^(7, it is manifest that the corresponding chord qq in the auxiliary circle will pass through the point / ; and if Cd be the seuii- diameter parallel toj^/ we shall have as before, pO . p'O : CcV :: qO' : q' 0' : AC 2 . But QO' . Q'O' = q0' . q'O', {Euclid, III. 35.) PO . D'O : CD 2 :: ^6> . p'O : Cd 2 , DO . D'O: pO . p'O :: CD 2 : Cd 2 . The same result may be shown to be true when the point is without the ellipse. Prop. XXX. If Q VQ' be any ordinate to the diameter CD, the circle described through the three points D, Q, Q' will intersect the ellipse in a fourth point, which depends only upon the position of P. Draw the ordinate DN, and produce it to meet the ellipse in D' ; then, since, if NT be the subtangent of either D orP', CT . CN^AC*, (Drop. IX.) therefore the tangents at P and P' will meet the major axis produced in the same point T. 62 CONIC SECTIONS. Draw PR parallel to TP', meeting the ellipse in R, and QQ' in 0; then if CD and CD' be drawn parallel re- spectively to TP and TP', meeting the ellipse in D and I)\ PO . OR : QO . OQ' :: CZT : OP 2 . (Prc> XXIX.) Bnt CD' = CD, since CP' = CP, Hence, by the converse of Euclid, III. Prop. 35, the point R is on the circle which passes through P. Q, Q'. Cor. When the point V is brought indefinitely near to P, ()§ 7 coincides with the tangent to the ellipse at P, and becomes also a tangent to the circle at P since Q and Q' are indefinitely near to each other. The circle therefore becomes the circle of curvature at the point P. Hence, if PR be drawn parallel to the tangent at P', or be equally inclined to the axis with PT, it will meet the ellipse in the point where the circle of curvature at P intersects the ellipse. PROBLEMS ON THE ELLIPSE. 1. In what position of P is the angle SPS' greatest ? 2. The latus rectum is a third proportional to the axis major and axis minor. 3. Construct on the axis minor as base, a rectangle which shall be to the triangle SLS' in the duplicate ratio of the major axis to the minor axis, L being the extremity of the latus rectum. 4. If a series of ellipses be described having the same major axis ; the tangents at the extremities of their latera recta will all meet the minor axis in the same point. 5. Find the locus of the centres of all the ellipses having the same focus, and their major axes of the same length, and touching a given straight line. G. Given the foci, it is required to describe an ellipse touching a given straight line. 7. If PT be a tangent to an ellipse, meeting the axis in T, and AP, A' P, be produced to meet the perpendicular to the major axis through T in Q and Q\ then QT = Q'T. 8. If the angle SBS' be a right angle, prove that CA 2 = 2CB\ 9. If CP be a semi-diameter, and AQO be drawn parallel to CP meeting the curve in Q, and GB produced in 0, then 2CP* = A0 . AQ. 10. If A B y CD, which are not parallel, make equal angles with either axis, the lines A C, BD } as also A D, BO, will make equal angles with either axis. 64 CONIC SECTIONS. 11. PSp is any focal chord. PA and pA are produced to meet the directrix in Q and q. Prove that the angle QSq is a right angle. 12. If a circle be described touching the axis major in one focus, and passing through one extremity of the axis minor ; A C will be a mean proportional between the diameter of this circle and PC. 13. HPQQ'P' be a chord of the auxiliary circle, and a circle be described on the minor axis as diameter, cutting the chord in Q andQ', then PQ . P'Q = CS\ 14. If PG be the normal at P, and GL be drawn at right angles to SP, then PL = I latus rectum. 15. The sum of the squares of the normals at the ex- tremities of conjugate diameters is constant. 16. If on the normal at P, PQ be taken equal to the semi- conjugate diameter CD, the locus of Q is a circle whose radius is AC — PC 17. Find the locus of the intersection of a pair of tangents at right angles to each other. 18. P is any point on an ellipse. To any point Q on the curve draw A Q, A'Q, meeting JVP in P and S, and prove that NR . NS = NP\ 19. If PG be a normal, and GL perpendicular to SP, the ratio of GL to P^Yis constant. 20. If NP produced meet the tangent at the extremity of the latus rectum in Q, then QN = PS. 21. In an ellipse the tangent at any point makes a greater angle with the focal distance than with the perpendicular on the directrix. 22. A diameter of an ellipse, parallel to the tangent at any point, meets the focal distances of the point, and from the points of intersection lines are drawn perpendicular to the focal distances. Prove that these lines intersect in the axis minor. 23. The subnormal is a third proportional to CT and PC. CONIC SECTIONS. 05 24. If PiV r be the ordinate of P, prove that NY : NY' :: PY.PY. (See fig. Prop. XV.) 25. If from C lines be drawn parallel and perpendicular to the tangent at P, they inclose a part of one of the focal distances of that point equal to the other. 26. If P be a fixed point on an ellipse, and QQ' an ordinate to GP, the circle QPQ' will meet the ellipse in a fixed point. 27. P is any point on an ellipse. Draw PP' parallel to the axis major, and through P' draw P' Q, P'Q' , making equal angles with the major axis. Join QQ' ; then QQ' is parallel to the tangent at P. 28. What parallelogram circumscribing an ellipse has the least area ? 29. When is the square of the sum of conjugate diameters least ? 30. Given the axes of an ellipse, and the position of one focus, and of one point in the curve, give a geometrical con- struction for finding the centre. 31. If lines drawn through any point of an ellipse to the extremities of any diameter meet the conjugate CD in M and N, then CM . CN = CD\ 32. If CP and GP) be conjugate, prove that (SP -AG) 2 + (SB - A C) 2 = SC\ 33. If CP and CD be conjugate, and PP, PD be joined, as also AD, A'P, these latter meeting in 0, then BDOPia a parallelogram. When is the area greatest ? 34. If PSp, QCq be two parallel chords through the focus and centre of an ellipse, prove that SP:Sp: CQ . Cq \iBC*zA&. 35. If the tangent at the vertex A cut any two conjugate diameters in T and t, then AT . At = BC' 1 . 3G. If the tangents at three points P, Q, B, intersect in B, Q r , P /} prove that PR t .P t Q. QP^PQ^B^.PB. F 66 CONIC SECTIONS. 37. If a circle be described touching SP, S'P produced, and the major axis of the ellipse, find the locus of the centre. 38. If from the extremities of the axes of an ellipse any four parallel lines be drawn, the points in which they cut the curve are the extremities of conjugate diameters. 39. If two equal and similar ellipses have a common centre, the points of intersection are at the extremities of diameters at right angles to one another. 40. If PSQ be a focal chord, and X the foot of the directrix, XP and XQ are equally inclined to the axis. 41. OP, OQ are tangents to an ellipse, and PQ is produced to meet the directrices in E, R\ prove that BP . B'P : PQ : B' Q :: OP 2 : OQ 2 . 42. XP Q is a common ordinate to the ellipse and auxiliary circle. PB, QR are normals at P and Q intersecting in B. The locus of B is a circle whose radius is A C + B C. 43. If the conjugate to OP meet SP, S'P, or these pro- duced in E, E' ; then SB = S' E', and the circles circum- scribing SUE, S'CE' are equal. 44. The locus of the middle points of all focal chords in an ellipse is a similar ellipse. 45. The circle described about the triangle SBS' will cut the minor axis in the centre of the circle of curvature at B. 46. The locus of the centre of the circle inscribed in the triangle SPS' is an ellipse. 47. If a circle be described intersecting an ellipse in four points, and chords be drawn through the points of intersection, diameters parallel to the chords will be equal. 48. An ellipse slides between two lines at right angles to each other, find the locus of its centre. 49. If from the focus S perpendiculars be drawn upon the conjugate diameters CP, CD, these perpendiculars produced backward will intersect CD and CP in the directrix. 50. Find the point at which the diameter of curvature is a mean proportional between the major and minor axes. CONIC SECTIONS. 67 51. The circle of curvature at a point, where the conjugate diameters are equal, meets the ellipse again at the extremity of the diameter. 52. The locus of the intersection of lines drawn from A, A at right angles to A P, A' P is an ellipse. 53. If a quadrilateral figure be inscribable in two ellipses whose major axes are parallel or perpendicular, any two of its opposite angles will be equal to two right angles. 54. If CJV, NP are the abscissa and ordinate of a point P on a circle whose centre is C, and NQ be taken equal to NP, and be inclined to it at a constant angle, the locus of Q is an ellipse. 55. If two ellipsis having the same major axes can be inscribed in a parallelogram, tiie foci will be on the corners of an equiangular parallelogram. 56. Two ellipses, whose major axes are equal, have a com- mon focus. Prove that they intersect in tw T o points only. 57. A circle described about the triangle SPS' cuts the minor axes in R on the opposite side to P. Prove that SR varies as the normal PG. 58. If r and R be the radii of the circles inscribed in and about the triangle SP S, prove that R . r varies as SP . S'P. 59. The circle -described upon PG as diameter cuts SP, S'P in iT and L. Prove that KL is bisected by PG, and is perpendicular to it. GO. If from S' a line be drawn parallel to SP, it will meet SY in the circumference of a circle. 61. ^and t are the points where the tangent at P meets the axes. CP is produced to meet in L the circle described about the triangle TCt ; prove that PL is half the chord of the circle of curvature at P in the direction of C, and that CP . CL is constant. 62. About the triangle PQR an ellipse is described, having its centre at the point where the lines drawn from P, Q, R, to the middle points of the opposite sides meet. CP, CQ, CR, are produced to meet the ellipse in P\ Q\ R\ Prove that F 2 68 CONIC SECTIONS. the tangents at P', Q', R' form a triangle similar to PQP, and four times as large. 63. Lines from Fand Y' perpendicular to the major axis cut the circles on SP, S'P as diameters in / and J. Prove that IS and JS' when produced, intersect BC in the same point. 64. If from the ends of any diameter chords be drawn to any point in the ellipse, the diameters parallel to these chords will be conjugate. 65. If T be the angle between tangents at the extremities of a focal chord, and the angle subtended by the chord at the other focus, then 2T + = 2 right angles. 66. The acute angles which SP, SQmake with the tangents are complementary. Prove that BC 2 is a mean proportional between the areas of the triangles SPS', SQS'. Also, show that the problem is impossible unless BC < CS. 67. A series of ellipses have their equal conjugate diameters of the same magnitude. One of these diameters is fixed and commou, while the other varies. The tangents drawn from any point in the fixed diameter produced will touch the ellipses in points situated on a circle. 68. If on the longer side of a rectangle as major axis an ellipse be described, passing through the intersection of the diagonals, and lines be drawn from any point of the ellipse exterior to the rectangle to the ends of the remote side, they will divide the major axis into segments, which are in geo- metric progression. 69. From any point P of an ellipse PQ is drawn at right angles to SP meeting the diameter conjugate to GP in Q. Prove that P Q varies inversely as the perpendicular from P on the major axis. 70. In an ellipse SQ and S'Q, drawn at right angles to a pair of conjugate diameters, intersect in Q. Prove that the locus of Q is a concentric ellipse. CHAPTER III. THE HYPERBOL A. 45. Def. The Hyperbola is the curve traced out by a point which moves in such a manner that its distance from a given fixed point continually hears the same ratio, greater them unity, to its distance from a given fixed line. (See Introduction.) Prop. I. The focus and directrix of a hyperbola being given, to find any number of points on the curve. Let 8 be the focus, and MX the directrix. Draw SX at right angles to the directrix, and divide SX in the point .4, so that SA may be to AX in the given fixed ratio, greater than unity ; then A is a point on the curve. On SX produced take a point A ', such that SA' : A'X:: SA : AX; then A' will also be a point on the curve. On the directrix take any point M\ and through S and M draw the line SYMY'. meeting AY and A' Y, drawn at right angles to A A', in the points Fand Y' ; On YY' as diameter describe a circle, and draw PMP parallel to A A', cutting the circle in the points Pand P' ; P and P 1 will be points on the hyperbola. '0 CONIC SECTIONS. Join PY, PY', SP; then since SY: YM :: 8 A : AX, {Euclid, VI. 2) and ST : Y 'M :: 5^' : A'X, {Euclid, VI. 2) .\ 57 : Of:: /ST': Y'M; or, alternately, SY : 5F:: FJf: F'Jf, and the angle 7P7' in a semicircle is a right angle, .-. PY bisects the angle SPM* .-. SP: PM:: SY: YM, :: SA : AX. So we may show that SP' :FM:: SY' : Y'M, * For, if not, make the angle YPm equal to YPS ; then SY : Ym :: SP : Pm. {Euclid, VI. 1.) and since, if PY bisect SPm, PY' being at right angles to P Y, also bisects the angle between MP and SP produced ; .-. SY' : Y'm :: SP : Pm, {Euclid, VI. A.) Hence £F : Fm :: ST : F'm, or ,?r : ST :: Ym : Y'm, .'. the points M and m coincide. CONIC SECTIONS. 71 ::SA:AX, .'. P and P' are points on tho curve. In the same way, by taking other points on the directrix, we may obtain as many more points on the curve as wo please. Cor. 1. Since, corresponding to every point P on the curve, there is a point P' situated in precisely the same manner witli respect to A'Y' as Pis with respect to A Y, it is clear that if we make A' S' equal to AS, and A' X' equal to AX, and draw X'M' at right angles to AX', the curve could be equally well described with 8' as focus and M' X' as directrix. The hyperbola is therefore symmetrical, not oidy with respect to the line A A', but also with respect to the line OG drawn through the middle point of Y Y' at right angles to and bisecting A A'. Cor. 2. The line OP produced will bisect the angle SPW between SP and S'P produced. Produce OP and S' S to meet in'Cr. Produce PM to meet X'M' in M', and draw OS' passing through the point M' ; then SP: PM:: S'P: PM', or, alternately, SP : S'P:: PM : PM'. (1) Again, SO : PM v. S'G : PM', or, alternately, SO : S' G :: PM : PM'. (2) .-. from (1) and (2) SP: S'P:: SG: S'G, .'. PG bisects the angle SPW. (Undid, VI. A.) It will be shown hereafter (Prop. IX.) that the normal to the hyperbola at the point P also bisects the angle SPIV. Hence the hyperbola and circle have the same tangent at the point P. The hyperbola will consequently touch all the infinite series of circles which can be described in the same manner as the one in the figure, by taking different points on ihe directrix. 72 CONIC SECTIONS. Prop. II. 46. If C be the middle point of A A', then CA is a mean proportional between CS and OX, or CS . CX = CA\ (See fig. Prop . in.) Since SA' : ■Ji. j\ '. '. SA : iX Alternately SA' : SA : : A'X : iJ, . *. >S4 — o-4 SA :: A'X- -AX: AX; or .44' : SA :: XX : A X, .*. ^x ./x : XX' : : SA : AX, or (7,4 : CX :: SA : AX. (1.)* Acjain, SA' : SA :: A'X : AX. .-. SA' + £4 : SA :: A'X+AX: 4X, or &S" : /SL4 : : AA': AX. Alternately, SS' : AA': : SA : AX, • or 0£ : CA : : £4 : AX. (20 Hence from (1) and (2) CA : CX : : CS . CA, • CA* = or. CS. or CA is a mean proportional between CS and OX Cor. Since the three lines CS, CA, CX, are proportional therefore, by the definition of duplicate ratio and Euclid, VI. 20 Cor., CS : CX :: CS 1 : Cy :\ (3 ) Prop. III. 47. If P be any point on the hyperbola, and S be the focus nearer to P; then S'P- SP=AA'. Since £P : PM n SA : A X, * N.B. The results (1), (2), (3), should he remembered, as they will frequently be referred to. CONIC SECTIONS. to and SA : AX : ^' : XX, . • . SP : PM :: .1.4' : X.I". So ,S"P : POT :: AA' : XX\ S'P - SP : PJ/' - PM :: .4.4' : XX'. But PJf - Pilf = JOT' = IT, .-. S'P- SP = .4.4'. (Pro]-). II.) Cor. By means of this property the hyperbola may be practically described, and the form of the curve determined. Let a rigid bar S'Q of any length have one end fastened at the focus S\ in such a manner that it is capable of turning freely round S' as a centre in the plane of the paper. At the other end of the bar let a string be fastened of such a length that when stretched along the bar it shall just reach to within a distance equal to .4.4' from the end >S" of the bar. 74 CONIC SECTIONS. If the loose end of the string be now fastened to the focus S, and the rod being initially placed in the position S' S, be made to revolve round S', while the string is kept constantly stretched by means of the point of a pencil at P, in contact with the bar; since S' P and SP are always increasing bv the same amount, viz. the length of the portion of the string that removes itself from the bar, between any two positions of P, the difference between S' P and SP will be constantly the same, and the point P will trace out the hyperbola. Another perfectly similar branch may be described in the same manner by making the bar revolve round S as centre. In this case S' P - SP will be equal to A A'. The curve, therefore, consists of two similar branches, which recede indefinitely both from the line AA', and also from the line BOB' drawn bisecting A A' at right angles. (See Jig. Prop. IV.) 48. If BC be taken of such a length that BC 2 = CS 2 - CA\ CONIC SECTIONS. lb and CB' be made equal to CB, then A A' and BB' are called respectively the Transverse and Conjugate Axes. The line BCB' does not meet the hyperbola, and the reason of its being introduced will be seen further on. If the conjugate and transverse axes are equal, the hyper- bola is said to be rectangular or equilateral. The property of the hyperbola, which we have just inves- tigated, viz. that the difference between SP and S P is constant, is sometimes taken as the definition of the curve. (See Chapter II. Art. 25.) Also as in the ellipse, if SL be the semi-latus rectum, it may be proved that SL . AC = BC\ Prop. IV. 49. The difference of the distances of any point from the foci of a hyperbola will be greater or less than A A', according as the point is on the concave or convex side of the curve. (1.) Let Q lie a point on the concave side of the hyperbola. Join SQ, S'Q, and let S'Q meet the curve in P; join SP; then since S'Q= S'P+ PQ, and SQ < SP -f PQ, 76 CONIC SECTIONS. ... S'Q- SQ> S'P- SP but S'P- SP= A A', ... S'Q- SQ> AA'. (2.) Let Q be a point on the convex side of the curve, nearer to S than S r ; join SQ, S'Q, and let SQ meet the curve in P; join S'P; then S'Q< S'P+ PQ, and SQ = SP 4- Pft ... s'Q-SQ< S'P- SP, but £'P- /S'P = A A ', ... S'Q-SQST £'P : SP, (Euclid, VI. 3.) PJP : PM, X'N; XK 80 CONIC SECTIONS. .-. S'T - ST y.S'T + ST::X'N- XN : X' N + XN ov2CT: 2CS::2GX: 2CN, or CT: GS :: CX : CX. .-. GT. CN=CS. CX, = CA\ {Prop. II.) 52. Def. The line PG, drawn at right angles to the tangent PT, is called the Normal to the hypjrbola at the point P. Prop. IX. If the normal to the hyperbola at the point P meet the transverse axis in the point G, and PX be the ordinate of the point P, then NG : NC :: BG l : A C\ Draw PMM' at right angles to the directrices, meeting them in M and M' , and produce S' P to W\ then since the angle TPG is a right angle, .*. the angle WPG = the complement of the angle S'PT, and the angle SPG = the complement of the angle SPT; CONIC SECTIONS. 81 but the angle S'PT = the angle SPT, .*. the angle WPG = the angle SPG, /. PG bisects the angle SPW, .-. S'G : SG :: &'P : SP, {Euclid, VI. A.) :: Z J Ji' : iW, :: X'AT: XX, .-. £'S : 6'X, :: OS 2 : CA 2 (Prop. II. <7or.) ... C'6' - tfAT : Otf :: OS 2 - CA 2 : CA 2 ; or XG : CX :: £C 2 : JO*. Plop. X. [f PX be the ordinate of any point P on the hyperbola, then PX 2 : AX . A'X :: EC 2 : AC 2 . YotNG : XC :: i/6' 2 : J6' 2 . And rectangles of the same altitude are to one another as their bases, (Euclid, VI. 1.) .-. TX . XG : TX . XC :: EC 2 : AC 2 ; or Pif 2 : TX . XC :: i/6' 2 : i('-. But 7'A' . CX = CX 2 - CT . CX, (Euclid, II. 2.) = CX 2 - CA 2 , (Prop. VIII.) = J if . A'X, (Euclid, II. 0) .-. PiV 2 : ^tiV . J^V :: EC : AC 1 . 1'iior. XI. If the normal at any point P of an hyperbola meet the transverse axis in G; then SG : SP :: OS : CA. G 82 CONIC SECTIONS. Troduce S'P to W; then since PG bisects the angle SPTV, (Prop. IX.) SG : S'G :: SP : S'P, ... SG : S'G - SG :: SP : S'P - SP, but S'P - SP = A A', (Prop. III.) and S'G - SG = SS', .-. SG : SS' :: SP : AA', or SG : SP :: SS' : A A', or SG : /ST :: CS : (M. Cor. Hence also, S'G : S'P :: CS : OA Prop. XII. 53. If from the foci S and 5" of an hyperbola S Y and /S" Y' are drawn at right angles to the tangent at P, then Y and F' are on the circumference of the circle described on A A ' as diameter, and 8T . AT' = PC 2 . Join SP, S'P, and produce SY to meet ^'P in W ; join CF; then since the angle #P F = the angle TTP F, (Prop. VI.) and the angle /S'FP = the angle WYP, and the side PF is common to the triangles SPY, WPY, .-. the triangle SPY = WPY in all respects, .-. SP = PW, and £F = WY, ... £'p - >SP = S'W, but £'P - SP = ^', (Prop. III.) .-. S'W = AA'. Again, .-. SG = CS', and SY= IVY, .-. #<7 : C£' :: #F : YW, .-. CF is parallel to S'W, .-. CF : SIT :: C£ :: S S\ CONIC SECTIONS. 83 .-. CY = iS'W = CA; so CY = CA. .'. Y and Y' are points on the circumference of the circle described upon A A' as diameter. Next, let $Fbe produced to meet this circle in Z, and join ZY'\ then since the angle Z YY' is a right angle .'. ZY passes through the centre C, .-. the angle. SCZ = the angle S'CY', . ' . SZ = S Y , .-. SY . S'Y' = SY . SZ, = SA . SA', (Euclid, III. 36 Cor.) = OT _ (7.4 2 , (j0mcZ*77, II. 0.) Cor,. If CD be drawn parallel to the tangent at P meeting S'P in E; then since CEP Y is a parallelogram, .-. PE - CY = v*C 84 CONIC SECTIONS. Prop. XIII. 54. To draw a pair of tangents to an hyperbola from an external point 0. Of the foci S and S', let S' be that which is nearer to 0. With centre S and radius equal to A A ' describe a circle. Join OS, OS' ; and let SO or S'O produced meet the circle in the point I. Now if be a point inside the circle M1M' it is evident that OS' is greater than 01; and if be outside the circle, since OS - OS' < A A' or SI, (Prop. IV.) .-. OS - OS' < OS - 01, .-. OS' > 01. With centre and radius OS' describe another circle cutting the former in the points M and M' , which it will always do since OS' is greater than 01. CONIC SECTIONS. 8.5 Join SM, SM', and produce them to meet the hyperbola in the points P and P\ Join OP, OP' ; these will be the tangents required. Join S'P, S'P' ; then since S'P - JSP = A A' = SM, .-. S'P = PM. And .-. S'P, PO = MP, PO, each to each, and 0S f/ = OM, .-. the angle OPS' = the angle OPM, .-. OP is the tangent at P. (Prop. VI.) So OP is the tangent at P'. The points of contact P and P' will be upon the same or opposite branches of the hyperbola according as SM and SM' require to be produced in the same or in opposite directions with respect to S, in order to intersect the hyperbola. Prop. XIV. If from a point a pair of tangents, OP, OP' be drawn to an hyperbola, then the angles winch OPand OP' subtend at either focus will be equal or supplementary according as the points of contact are in the same or opposite branches of the hyperbola. Let the points P and P' be on opposite branches of the hyperbola. Join PS, S'P; SP', S'P'. Produce PS to M, making PM equal to PS'. Also from PS cut off a part PM' equal to PS'. Join OM, OM'; OS, OS'. Then since OP, PS' = OP, PM, each to each, and the angle OPS' = the angle OPM, {Prop. VI.) .-. OS' = OM, and the angle S' P = the angle OM P. 86 CONIC SECTIONS. So OS' = 031', and the angle OS'P' = the angle 031' P', .-. OM = 031'. Again, v SM = S'P - SP = AA', and SM' =. SP' - ST' = A A', .-. SM = #Jf'. And •.• 6>£, #Jf = OS, SM', each to each, and OM = 031', .-. the angle OSM = the angle OSM', and the angle OMS = the angle OM' S. But 0£Jf is the supplement of OSP, and OJ/'S is the supplement of OM'P', .'. S M' is the supplement of OSP, and MP the supplement of 031' P'. But 03IP = OS'P, and OM'P' = OS'P', .'. S' P is the supplement of OS'P. CONIC SECTIONS. »t"T Hence the angles which OP and OP subtend either at S or S' are supplementary. In a similar manner if P and P' are on the same branch of the hyperbola, the angles subtended either at S or >S" may be shown to be equal. Pkop. XV. 55. If the tangent at any point P of an hyperbola meet the conjugate axis in the point t, and Pn be drawn at right angles to GB\ then On . Ct = PC 2 . Draw Py Lt right angles to C A ; then Ct : CT v. PN : NT, .-. Ct : PN v. CT : NT, .'. Ct . Cn : PN 2 :: CT . CN : CN . NT; or Ct . Cn : CT . CN :: PN' : ON . NT, :: PC' 2 : AC*. (Prop. X.) But CT . Ci\T = .K'*, .-. a . Cfo = BC\ 88 CONIC SECTIONS. 50. The proofs that we have given up to this point of the properties of the hyperbola are closely analogous to the corresponding propositions in the ellipse. The remaining properties of the hyperbola are more conveniently investigated by means of its relation to certain lines, which we shall presently define, called Asymptotes, in the same manner as many of the properties of the ellipse were deduced from those of the auxiliary circle. Def. The hyperbola described (see Jig. Prop. XIV.) with C as centre, and BB' as transverse axis, and A A' as con- jugate axis, is called the Conjugate Hyperbola. Its foci, which will be on the line BOB', will evidently be at the same distance from C as those of the original hyperbola, since CS 2 = CA 2 + OB 2 . Vmv. XVI. If through any point R or either of the diagonals of the rectangle formed by drawing tangents to the hyperbola and its conjugate at the vertices, A, A', B, B', two ordinates RPN, RDM, be drawn at right angles to A A' and BB', and meeting either the hyperbola or its conjugate in the points P and D ; then RN 2 ~ P^V 2 = BC 2 , and RM 2 ^ DM 2 = AC 2 . Let R be a point on the diagonal 0' CO ; then RN 2 : CN 2 :: A 2 : AC 2 , :: BC 2 : AC 2 , and PA"" 2 : CN 2 - CA 2 :: BC 2 : AC 2 ; (Prop. X.) ... BN 2 - PN 2 : CA 2 :: BC 2 : AC 2 ; .-. RN 2 - PN 2 = BC 2 . Again, RM 2 : CM 2 :: AC 2 : BC\ and DM* : CM* - CB 2 :: AC 2 : BC 2 ; (Prop. X.) .-. RM 2 - DM 2 : BC 2 :: AC 2 : BC 2 ; .-. RM 2 - DM 2 = AC 2 . CONIC SECTIONS. In exactly the same manner, if NR had been produced to meet the conjugate hyperbola in P, and MR had been produced to meet the original hyperbola in D, we should have had, and DM' 2 - RM 2 = A C\ Con. If RP be produced to meet the hyperbola in p, and the other asymptote in r ; then EN 2 - PN* = RP . Pr ; (7?mc7/J, II. 5.) ..RP. Pr = BC\ Hence as RPN is further removed from A, and the line Pr consequently increases, since the rectangle contained by RP and Pr remains constant, RP must diminish, and by taking R sufficiently far from C, RP may be made less than any assignable magnitude. The line CR, therefore, con- tinually approaches nearer and nearer to the hyperbola, though it never actually reaches it. 90 CONIC SECTIONS. On account of this property, CB is called an Asymptote to the hyperbola. So also if P be the point where NB produced meets the conjugate hyperbola, we shall have BP . Pr = PC 2 ; and therefore CB is also the asymptote to the conjugate hyperbola. In the same manner it may be shown that the other diameter oCo' of the rectangle 00' is an asymptote to both hyperbolas. Prop. XVII. 57. If E be the point where the asymptote meets the directrix ; then CE=AC. CONIC SECTIONS. 01 For by similar triangles, CE: CO:: OX: CA, :: CA: CS. (Prop. II.) But CO 2 = CA' 2 + CB 2 = CS' 2 ; .\CO=CS; .-. CE=AC. Con. If SE be joined, since CE 2 = CA 2 = CS. CX, .'. the angle CES is a right angle. [Euclid, VI. 8, Cor.) Prop. XVIII. If from any point R in one of the asymptotes to an hyper- bola ordinates RPN, RDM be drawn to the hyperbola and its conjugate respectively, and PD be joined, ED will be parallel to the other asymptote. For EN 2 : EM 2 :: BC 2 : AC 2 ; and RN 2 - PN 2 : RIP - DM 2 ::BC 2 :A C'\ (Prop. XVI.) .-. PN 2 :DM 2 ::BC 2 :AC l ; :: EN 2 : RM\ .-. PN : DM:: EN : RM; .-. PD is parallel to MN. (Euclid, VI. 2.) Also CJV: CM::AC:BC } .-. ifiV is parallel to AB; and 6L1 :.lo:: OB: £0', .*. AB is parallel to 00'. Hence PD is parallel to 00'. Cor. So also if R and D be the points where JS T R and JO produced meet respectively the conjugate and the original hyperbola, PD will be still parallel to 00'. C9 CONIC SECTIONS. Prop. XIX. 58. If through any two points Q and Q' of an hyperbola a line R Q Q'R' be drawn in any direction meeting the asymptotes in R and R' ; then will RQ = R'Q. Through Q and Q' draw the ordinates UQW, U'Q'W; meeting the asymptotes in U, W, U, W ; then by similar triangles, QR : QU :: Q'R : Q'U', and QR ' : QW :: g'^' : Q'W ; .'. compounding QR. QR' : QU. QW :: Q f R . Q' R' : Q'U' . Q'W. But QU . QJF ~ RC 2 = Q' U' . Q' JV, {Prop. XVI. Cor) .-. QR. QR' = Q'R . O'TT; CONIC SECTIONS. 93 but QR . QR' = QR . QQ' + QR . Q' R\ and Q'R . Q'R' = Q'R' . Q Q' + #£ . <2'i2' ; • .-. QR. QQ' = Q'R. QQ'. .'. QR^Q'R'. Cok. 1. If RQQ'R' move parallel to itself until tlie points Q and Q' coincide, the line RQR' will ultimately assume the position LPl, and will become a tangent to the hyperbola at P. Hence, since R Q is always equal to R' Q', LP=Pl, or the tangent LPl is bisected at the point of contact P. Con. 2. If CPhe produced to meet RR' in V, then since RV : VR' :: ZP : PI, .". A I = V Jx j and 72 tf = Q'R' } .'.QV=Q'V. Hence, if a series of parallel chords be drawn in an hyper- bola, their middle points will all be in the line drawn through the centre and the point where the tangent parallel to the chords meets the hyperbola. Dee. A line PGP' drawn through the centre, and meeting the hyperbola in P and P, is called a Diameter. A diameter consequently bisects all chords drawn parallel to the tangents at its extremities. Prop. XX. 59. If through any point Q of an hyperbola a line RQr be drawn in any direction meeting the asymptotes in R and r, and LPl be the tangent drawn parallel to R Qr ; then RQ . Qr = PL\ Through P and Q draw the ordinates EPc, UQ IV, meeting the asymptote in L\ c, U, IV ; then by similar triangles, 94 CONIC SECTIONS. QR : QU Qr: QW .-. QR . Qr : QU . QW PL : PP, : PI : Pc; PL .PI : PE.Pe; but QU . QW=BC 2 = PE . Pc, (Prop. XVI. Cor.) .-. QR. Qr = PL. PI, = PL 2 . (Prop. XIX. Cor. 1.) Cor. If Qq be produced to meet the conjugate hyperbola in Q', q, we may show that Q'R. Q'r = PL 2 , and also, as in Proposition XIX., that Q'R = qr, •'• QQ' = ¥1- Hence if a line be drawn in any direction meeting both the hyperbolas, the portions intercepted between the hyperbola and its conjugate will be equal. CONIC SECTIONS. 9: Peop. XXI. GO. If from any point P of an hyperbola, P//and PA' be drawn parallel to the asymptotes, meeting them in H and K respectively ; then 4 . PH : PK = CS 2 . Draw the ordinate IIP If r meeting the asymptotes in R and r ; then by similar triangles, PH : PR and PK : Pr .\PH. PK : PR.Pr Co : Oo y CO : Oo, CO 2 : Oo 2 , CS 2 : ±BC 2 . But PR . Pr = B C 2 } .4.PH.PK= CS 2 . 06 CONIC SECTIONS. Prop. XXII. If the tangent at any point P of an hyperbola meet the asymptotes in L and / ; then the area of the triangle L CI is equal to the rectangle contained by A C and B C. Draw PH and PK parallel to the asymptotes meeting them in II and K\ then since CL : CH :: LI : PI, and LI = 2 PI, (Prop. XIX. Cor. 1.) .-. CL = 2CII = 2PK; so CI = 2 CK - 2 PH, .-. CL. Cl = 4.PH . PK= CS\ (Prop. XXI.) = CO . Co, .-. CL: CO :: Co: CI, .' . the triangles L CI, Co have the angle at C common and the sides about those angles reciprocally proportional .'. the triangle LCI = the triangle OCo, = AC.A0. = AC . BC Puop. XXIII. 01. If from any point R in the asymptote of an hyperbola two ordinates UPN?m& II D M be drawn to the hyperbola and its conjugate respectively, then the tangents at P and D will be parallel respectively to Cl) and CP. Join PI), meeting CB in II; then since PL is parallel to oo , (Prop. XVIII.) the tangents at P and D will each meet CB produced in the same point L. (Prop. XXII.) Produce LP and LB to meet the other asymptotes in / and V ; then since CL . Cl = CS 2 = CL . Cl', (Prop. XXII.) CONIC SECTIONS. 97 .-. CI = Cl\ .'. IC : CV :: IP : PL, .'. CP is parallel to the tangent at D. Also I'D : PL :: V C : CI, .'. CP is parallel to the tangent at P. The lines CP and CP are called Conjugate Piamders, since each of these lines is parallel to the tangent at the extremity of the other. Prop. XXIV. If CP and CP be semi-conjugate diameters in the hyper- bola ; then CP 2 ■-, CP 2 = CA 2 --, CP 2 . Draw the ordinates NPP,MPR meeting the asymptote in the point R {Prop. XXIII.) ; then 98 CONIC SECTIONS. CB 2 - CP 2 = MB 2 - NP\ BG\ {Prop. XVI.) CB 2 + BC 2 ; . CD 2 + AC 2 , CD 2 + AC 2 ; AC 2 <-, BC 2 . .-. CB 2 so CB 2 .-. CP 2 + BC 2 or CP 2 -, CD 2 Peop. XXV. 62. The area of any parallelogram formed by drawing tan- gents to the hyperbola and its conjugate at the extremities P, P\ D, D' of a pair of conjugate diameters is equal to the rectangle contained by the axes. Let LI, L'V be the parallelogram formed by drawing tangents at the extremities P, P\ D, D', of airy pair of conjugate diameters. The points L, L', I, V , will {Prop. XXIII.) be on the asymptotes. CONIC SECTIONS. 09 Now the parallelogram LL' = 4 parallelogram CL, = 4 triangle L CI, = ±AC.BC, (Prop. XXII.) = AA'.BB\ Cor. If PF be drawn perpendicular to CD, then PF. CD = AC. BC. Also, if the normal PC meet the transverse axis in G, as in the ellipse BF.PG = BC 2 . 63. Def. The line QV drawn from any point Q of the hyperbola parallel to the tangent at any point P, and meeting CP produced in V, is called an Ordinate to the diameter CP. Prop. XXVI. If Q V be an ordinate to the diameter P' CP, and CD be conjugate to CP ; then QV 2 : PV . P'V :: CD 2 : CP\ Produce VQ to meet the asymptotes in R and r ; and let the tangent at P meet the asymptotes in L and I ; then BV 2 : PL 2 :: CV 2 : CP 2 , .-. BV 2 - PL 2 : PL 1 :: CV 2 - CP 2 : <7P 2 . But RQ. Qr = PL 2 , (Prop. XX.) .-. BV 2 - QV* = PL 2 , or i2 V 2 - PL 2 = Q V 2 . And CV 2 - CP 2 = P V . B' V, (Euclid II. 6.) .-. QV 2 : PZ 2 :: PF . P r F : CP 2 . Alternately, § F 2 : PF . P7:: BL 2 : CP 2 . But since PD is a parallelogram, (Prop. XXIII.) .-. PL = CD. Hence QV 2 : PV . P'F :: OZ) 2 : (7P 2 . Cor. If VQ be produced to meet the conjugate hyperbola in Q' , then since Q'R . Q'r = PL 2 , (Prop. XX. Cor.) ... <3'F 2 -PF 2 = PP 2 . Hence Q' V 2 : CV 2 + CP 2 :: CP 2 : CP 2 . H 2 100 CONIC SECTIONS. Prop. XXVII. 04. If Q V be an ordinate to the diameter P V, and the tangent at Q meet CP in the point T ; then CV. CT = CP\ Draw the tangent LPl meeting the asymptotes in the points L, I ; also let the tangent at Q meet the asymptotes in P, r. Draw 11 K t rk, parallel to Q V meeting CP in K, k. Xow since the triangles PCr, LCI are equal, (Prop. XXII.) and have the angle at C common, .-. CP : CL :: CI : Cr. (Euclid, VI. 15.) But CP : CL :: CK : CP, CONIC SECTIONS. lUl and CI : Cr : : CP : Ck, .'. CK : CP :: CP : Ck, .\CK . Ck = CP 2 . - Again, produce RK and QV to meet the asymptote CI in B' and q ; then Since Br is bisected in Q, (Prop. XIX. Cor. 1.) .*. JKV is bisected in q, and iZZ = B'K, {Prop. XIX. GW. 2.) . ' . Kq is parallel to Rr, .'. CT : G'A" :: 6V : Cq, :: CT : OF, .-. CV . CT = CK. Ck = CP\ Cor. 1. Conversely, if QVbe an ordinate to PY, and CV . CT = CP 2 , then Q T is the tangent at Q. Cor. 2. Hence also, if PR' meet the curve in £7 and U', and & Z7, k V be drawn, since CK . Ck = CP\ .-. k £7" and k V are tangents to the hvnerbola at U and V. Pkop. XXVIII 65. If two chords of a hyperbola intersect one another, tin 1 rectangles contained by their segments are proportional to the squares of the diameters parallel to them. Let QOq be any chord drawn through the point 0, and let CD be drawn parallel to it, meeting the conjugate hyperbola in D. Produce Qq to meet the asymptotes in P and r ; and draw the diameter CPV, bisecting both Qq and Br in 7". {Prop. XIX. Cor. 2.) Also draw the tangent LPl parallel to Qq, meeting the asymptotes in L and I. 102 CONIC SECTIONS. Now since Qq is divided equally in J 7 " and unequally in 0, ... QO . Oq = Q V 2 - V 2 ; (Euclid, [I. 5.) so also RO.Or = B V 2 - V 2 ; {Euclid, II. 5.) ... no . Or -Qo .Oq = n v 2 - q v\ = RQ . Qr (Euclid, II. 5.) = PL 2 , (Prop. XX.) .-. QO .Oq = EO . Or -PL 2 . Again, through and P draw EOc, UPW, at right angles to the axis meeting the asymptotes in E, e, U,W\ then PO : OE : and rO : Oc : '. RO . rO : OE . Oc : PL : PU, ' PI : PJF, PL 2 : PZ7, P7F; CONIC SECTIONS. 103 but PU. PW =■ PC 2 , (Prop. XVI.) and PL* = CD' 2 , (Prop. XXIII.) .*, RO . rO : OP . Oe :: CD 2 : J?C a , ovBO.rO : CD 2 :: OP . Oe : BC 2 . PO. rO - PL 2 : G'P 2 :: OP . Oe - BC 2 : BC 2 , or QO . Oq : CD 2 :: OE . Oe - BC 2 : BC 2 . In the same manner if through another chord Q' Oq he drawn, and CD' be drawn parallel to it, meeting the con- jugate hyperbola in D' , we shall have Q'O . Oq' : CD' 2 :: OE . Oc - BC 2 : BC 2 . Hence QO . Oq : Q'O . Oq' :: CD 2 : CD' 2 . Cor. The same result may be shown to be true when the point is outside the hyperbola. Moreover, it is not necessary that the chords should be drawn meeting one branch only of the hyperbola or the same branch. The proportion still holds good when one or both of the chords meet both branches of the hyperbola, or when the chords are drawn in different branches. 66. Def. If with a point on the normal at P as centre, and OP as radius, a circle be described touching the hyperbola at P, and cutting it in Q ; then when the point Q is made to approach indefinitely near to P, the circle is called the Circle of Curvature at the point P. Prop. XXIX. If Pi?" be the chord of the circle of curvature at the point P of a hyperbola, which passes through the centre ; then PH . CP = 2 CD 2 . Let PT be the tangent, and PG the normal at the point P. With centre 0, and radius P, describe a circle cutting the hyperbola in the point Q. Draw B Q W parallel to CP, meeting the circle in W, and TP produced in B. Also, draw Q V parallel to PB, meeting the diameter PP' in V\ then, since PP touches the circle at P, 104 CONIC SECTIONS. .*. BQ or PV But QV 2 : PV .-. PV.BW: PV . P'V :: CD 2 orPW : P'V :: CD 2 BW = PP 2 , (-EtoeWtf, HI. 36.) JSJF = QV 2 . CP\ (Prop. XXVI.) CP 2 , CP 2 . P'V v. CD 2 ISTow, when the circle becomes the circle of curvature at P, the points B and Q move up to, and coincide with P, and the lines B TV and PH become equal, while P' V becomes equal to PP' , or 2 CP. Hence, PH : 2CP :: CP 2 : CP\ .-. PZT . CP : 2CP 2 :: 2 CD 2 : 2 CP 2 , .-. PH . CP = 2 CD 2 . Prop. XXX. If PC be the diameter of the circle of curvature at the point P of the hyperbola, and PF be drawn at light angles to CD; then PU . PF = 2 CD 2 . CONIC SECTIONS. 105 Since the triangle PR U is similar to the triangle PFC, ..PIT: PR:: CP : PF, .'.PU . PF=PR: CP, = 2 CD\ {Prop. XXIX.) Prop. XXXI. If PI be the chord of the circle of curvature through the focus of the hyperbola ; then PI . A C = 2 CP 2 . Let S'P meet CP in E ; then, since the triangles PIU and PEF are similar, .-. PI : PU :: PF : PE. But PE = AC, (Prop. XII. Cor.) .'. PI : PU :: PJF : AC, .-. PI . AC = PU . PF, = 2 Ci) 2 . (Prop. XXX.) The point where the circle of curvature intersects the hyperbola may be determined as in the case of the ellipse. Pkop. XXXII. 67. If Pbe any point on the hyperbola, and CD be conju- gate to CP ; then sp . s'P= cm Draw PIP parallel to the asymptote CE meeting the directrices in / and /', and CB' in U. Let the ord mates, NP, MD meet the asymptote in P, and draw P W perpendicular to the directrix ; then by similar triangles, PI : PW :: CE : CX, :: CA : CX. (Prop. XVII.) 106 CONIC SECTIONS. But SP:PW :: SA : AX, :: CA : CX. .'. SP = PI; so S'P = PI\ .-. sp. s'p=pi.pi', = UP 2 - UP, = CE 2 - CE 2 , = CE 2 - GA\ {Prop. XVII.) But CE 2 - CD 2 = EM 2 - DM 2 , = CA 2 , (Prop. XVI.) .-. CE 2 -CA 2 = CD 2 , Hence SP. S'P= CD 2 . PROBLEMS OX THE HYPERBOLA. 1. The locus of the centre of a circle touching two given circles is an hyperbola or ellipse. 2. If on the portion of any tangent intercepted between the tangents at the vertices a circle be described, it will pass through the foci. 3. In an hyperbola the tangents at the vertices will meet the asymptotes in the circumference of the circle described on SS' as diameter. 4. If from a point P in an hyperbola PII' be drawn parallel to the transverse axis meeting the asymptotes in / and /', then PI . PI' = A C\ 5. If a circle be inscribed in the triangle SPS', the locus of its centre is the tangent at the vertex. 6. If PN be the ordinate of the point P, and NQ a tangent to the circle described on the transverse axis as diameter, and PM be drawn parallel to Q G meeting the axis in M, then MN =BC. 7. If PA" be the ordinate of a point P, and KQ be drawn parallel to ^.Pto meet G'Pin Q, then AQ is parallel to the tangent at P. 8. If an hyperbola and an ellipse have the same foci, they cut one another at right angles. 9. If the tangent at P intersect the tangents at the vertices in P, r, and the tangent at P' intersect them in P' } r', then AR. Ar = AR'.Ar\ 108 CONIC SECTIONS. 10. If any two tangents be drawn to an hyperbola, the lines joining the points where they intersect the asymptotes will be parallel. 11. The perpendicular drawn from the focus to the asymp- totes of an hyperbola is equal to the semi-conjugate axis. 12. If the asymptotes meet the tangent at the vertex in 0, and the directrix in E; then A E is parallel to SO. 13. In a rectangular hyperbola conjugate diameters are equal to one another. 14. In a rectangular hyperbola the normal PG is equal to CP. 15. The lines drawn from any point in a rectangular hyperbola to the extremities of a diameter make equal angles with the asymptotes. 16. Prove that the asymptotes to an hyperbola bisect the lines joining the extremities of conjugate diameters. 17. A line drawn through one of the vertices of an hyperbola and terminated by two lines drawn through the other vertex parallel to the asymptotes will be bisected at the other point where it cuts the hyperbola. 18. P is any point on an hyperbola, and P' a point on the conjugate hyperbola. If CP and CP' be conjugate, prove that S'P' -SP = AC-BC, S and /S" being the interior foci. 19. If OP and CD be conjugate, and through C a line be drawn parallel to either focal distance of P, the perpendicular from I) upon this line is equal to B C. 20. Given a pair of conjugate diameters, find the principal axes. 21. If Q be a point on the conjugate axis of a rectangular hyperbola, and Q P be drawn parallel to the transverse axis meeting the curve in P, then PQ = AQ. 22. In a rectangular hyperbola the focal chords drawn parallel to conjugate diameters are equal. CONIC SECTIONS. 109 23. If in an equilateral hyperbola C Y be drawn at right angles to the tangent at P, and A Yhe joined, the triangles PC A, CA I^are similar. 24. The radius of the circle which touches an hyperbola and its asymptotes, is equal to that part of the latus rectum produced which is intercepted between the curve and the asymptotes. 25. If QQ' be any chord of an hyperbola, and CP the diameter corresponding to it, and QH, PK, Q' H' be drawn parallel to one asymptote meeting the other in H, K and //', then CII . CH' = CK\ 26. If the chord RPP'R' intersect the hyperbola in the points P, P\ and the asymptotes in R, R ' ; and PK be drawn parallel to CP', and P 'K to CP ; then RK=P'K', and R'K = PK. 27. If A A' be any diameter of a circle, and PXQ an ordinate to it, then the locus of the intersections of AP, A'Q is a rectangular hyperbola. 28. If two concentric rectangular hyperbolas be described, the axes of one being the asymptotes of the other, they will intersect at right angles. 29. If any chord AP through the vertex be divided in Q, so that AQ : QP :: AC' 1 : BC\ and QN be drawn to the foot of the ordinate PN, prove that a straight line drawn at right angles to QN from Q cuts the transverse axis in the same ratio. 30. Prove that the curve which trisects the arcs of all segments of a circle described upon a given base is an hyper- bola. 31. If SVs, TVt be two tangents cutting one asymptote in #, T, and the other in s, t, prove that VS : Vs :: Vt : VT. 32. If from the exterior focus of an hyperbola a circle be described with radius equal to PC, and tangents be drawn to it from any point in the hyperbola, the line joining the points of contact will touch the circle described on the transverse axis as diameter. 110 CONIC SECTIONS. 33. Circles are drawn touching the straight line AB in a fixed point C; and from the fixed points A, B tangents are drawn to these circles. The locus of their intersection is an ellipse or hyperbola. Distinguish between the two cases. 34 PP' is a double ordinate in an ellipse. AP, A'P' are produced to meet in Q. Prove that the locus of Q is an hyperbola with the same axes as the ellipse. 35. If the tangent at P intersect the asymptotes in L and /, and PG be the normal at P, then the angle L Gl is a right angle. 36. If an ellipse, a parabola, and a hyperbola, have a common tangent, and the same curvature at the vertex, the ellipse will be entirely within the parabola, and the parabola entirely within the hyperbola. 37. The chord BBB'B' of an hyperbola intersects the asymptotes in B and B\ From the point B a tangent BQ is drawn meeting the hyperbola in Q. If PH, QK, P'H' be drawn parallel to one asymptote meeting the other in the points H, K, H' ; then PH. 4 P'H' = 2QK. 38. If through P,P' on an hyperbola lines be drawn parallel to the asymptotes forming a parallelogram, of which PP is one diagonal, the other diagonal will pass through the centre. 39. If Pbe the middle point of a line EF which moves so as to cut off a constant area from the corner of a rectangle, its locus is an equilateral hyperbola. 40. PM, PN are drawn parallel to the asymptotes ON, CM, and an ellipse is constructed having CJSf, CM for serni- conjugate diameters. If CP cut the ellipse in Q, the tangents at Q and P to the ellipse and hyperbola are parallel. 41. If a circle be described through any point P of a given hyperbola and the extremities A, A' of the transverse axis, and JSfP be produced to meet the circle in Q ; prove that Q traces out an hyperbola whose conjugate axis is a third pro- portional to the conjugate and transverse axes of the original hyperbola. CONIC SECTIONS. 11 1 42. If lines "be drawn from any point of a rectangular hyperbola to the extremities of a diameter, the difference between the angles which they make with the diameter will be equal to the angle which this diameter makes with its conjugate. 43. If between a rectangular hyperbola and its asymptotes any number of concentric elliptic quadrants be inscribed, the rectangle contained by their axes will be constant. 44. In the rectangular hyperbola if GP be produced to Q so that PQ = GP, and QO be drawn at right angles to CQ to intersect the normal in 0, is the centre of curvature at P. 45. With two conjugate diameters of an ellipse as asymp- totes a pair of conjugate hyperbolas are constructed ; prove that if one hyperbola touch the ellipse the other will do so likewise; prove also that the diameters drawn through the points of contact are conjugate to each other. 46. If a pair of conjugate diameters of an ellipse when produced be asymptotes to an hyperbola, the points of the hyperbola at which a tangent to the hyperbola will also be a tangent to the ellipse, lie in an ellipse similar to the given one. 47. In the rectangular hyperbola the radius of curvature at P is to the radius of curvature at P' in the triplicate ratio of GP to GP'. 48. OP, OQ are tangents to an ellipse at P and Q, and asymptotes to an hyperbola. Show that a pair of their common chords is parallel to PQ. One of these chords being PS, prove that if PR touches the hvperbola at P, QS touches it at S; also if PS, QB meet in U, U bisects PQ. 49. The base of the triangle AEG remains fixed, while the vertex C moves in an equilateral hyperbola passing through A and B. If P, Q be the points in which A G, BG meet the circle described on AB as diameter, the inter- section of A Q, BP is on the other branch of the hyperbola. CHAPTER IV. THE SECTIONS OF THE CONE. 68. Def. If two indefinite straight lines IOT, BOD', inter- sect one another at a point 0, and one of them IOT remain fixed while the other DOB' revolves round it in such a manner that its inclination to IOT is the same in all positions, the surface generated by B OD' will be a Bight Cone. The line IOT is called the Axis, and the point the Vertex of the Cone. It now remains for us to show (see Introduction) that the curve formed by the intersection of this surface with a plane is in general one of the three curves whose properties we have been investigating, and to consider under what circumstances it will be the Parabola, Ellipse, or Hyperbola. If the cutting plane pass through the vertex of the cone as KOK', and intersect the cone again at all, it will in general cut it in two straight lines as OK, OK', which will represent two positions of the generating line. The inclination of these lines to each other will depend upon the inclination of the cutting plane to the axis of the cone, and will be greatest when this plane passes through the axis, in which case it will be double the constant angle between the axis and the generating line. If the cutting plane pass through a generating line dod' and be perpendicular to the plane containing this line and the axis, it will simply touch the cone along this line. CONIC SECTIONS. 113 Should the cutting plane not pass through the vertex, and bo at right angles to the axis of the cone, the section will evidently be a circle. In any other case the section will, as we proceed to show, be a Parabola, Ellipse, or Hyperbola. Whatever be the position of the cutting plane with respect to the cone, we can always suppose a plane drawn through the axis of the cone at right angles to it ; and it will be convenient to have this latter plane represented by the plane of the paper as DOcl. The cutting plane will therefore always be taken at right angles to the plane DOd of the paper. 114 CONIC SECTIONS. Prop. I. 69. The curve formed by the intersection of the surface of a right cone with a plane (which neither passes through its vertex nor is at right angles to its axis) will be a Parabola, Ellipse, or Hyperbola, according as the inclination of the cutting plane to the axis of the cone is equal to, greater, or less than the constant angle which the generating line forms with the axis. Let the plane of the paper represent the plane drawn through the axis 1 01' of the cone at right angles to the cutting plane ; and let it intersect the surface of the cone in the two generating lines OD, Od. Let the cutting plane intersect the surface of the cone in the curve PA, and the plane of the paper in the line A NIL The curve will evidently be symmetrical with respect to this line. On All take any point N, and through JV draw a plane perpendicular to the axis meeting the surface of the cone in the circle RPr, and the cutting plane in the line PN, which will be at right angles to the plane of the paper and to AN. Let a sphere be inscribed in the cone touching the cone in the circle EQe and the cutting plane in the point 8, and let the plane EQe intersect the cutting plane in the line XM, which will be at right angles to the plane of the paper, and therefore parallel to PN. Draw PM perpendicular to XM, and join PS, PO, and let PO meet the circle EQe in the point Q* Then since PS and PQ are both tangents to the sphere, .-. PS = PQ. But PQ - RE, .-. PS = BE. * N.B. In the figure, to avoid confusion, that part of the section which lies above the plane of the paper is alone represented. CONIC SECTIONS. 115 II I But RE : XN :: AE : AX, (Euclid, VI. 2.) and AE = AS, .'. BE : XN v. AS : AX, .'. SP : PM :: AS : AX, . \ the curve PA is either a Parabola, Ellipse, or Hyperbola, ■whose focus is # and directrix A' J/. Again, let ^iZ" meet the axis 01 in F. Then the angle AFO will be the inclination of the cutting plane to the axis. (1) Let the angle AFO = the angle FOd ; then AJIis parallel to Od, .-. the angle AXE = the angle OeE = the angle -4jELY, .-. AS = AX, .-. the curve AP will be a Parabola. I 2 116 CONIC SECTIONS. CONIC SECTIONS. 117 (2) Let the angle AFO be > FOd; then the complement FXE is < the complement OEe, .*. the angle AXE is < A EX, .'. AE is < AX, or A Sis < AX, .'. the curve ^4 Pis an Ellipse. Since the angles UFO, FOd are together less than I1FO, OF A, i.e. than two right angles, the lines AH and Oe may be produced to meet in A'. If another sphere be described touching the cone in the circle E' Q' e and the cutting plane in the point S' ; and the line X' M' denote the intersection of the plane E'Q'ef with the cutting plane, and PM' be drawn at right angles to this line, it can easily be shown that S'P : PM' :: 8' A' : A X '. Hence S' and X'M represent respectively the other focus and directrix of the ellipse. Also if BCbe the semi-axis minor, and through the centre G a line TJGTJ' be drawn parallel to Ee meeting OB, Od in U and U', then it is evident that BC 2 = CU . czr. (3) Let the angle AFO be < FOd ; then the angle AXE is > the angle A EX, .-. AEis > AX, .-. AS is > AX, .'. the curve PA is an Hypcrhola. Since the angles AFO, FOd' are less than the two FOd, FOd', i.e. than two right angles, the lines FA and d may be produced to meet in A' . In this case the cutting plane will intersect the other half of the cone, and if any point P' be taken on this part of the curve, and P' M be drawn at right angles to XM, it can be shown as before that SP' : P'Jf :: S'A :: AX. 118 CONIC SECTIONS. CONIC SECTIONS. 119 The intersection of the catting plane therefore with this portion of the cone will be the other branch of the hyperbola. Also if another sphere be described touching the upper portion of the cone in E'Q'e', and the cutting plane in S', and the line X' M' denote the intersection of the plane E'Q'e' with the cutting plane, and P'M' be drawn at right angles to this line, it can be easily shown that S'M' : P'M' :: S'A' : A' X' . Hence, S' and X ' M' will represent respectively the other focus and directrix of the hyperbola. Cor. 1. In this last case, i.e. when the section is an hyper- bola, if a plane OKL be drawn through the vertex of the cone parallel to the cutting plane, meeting the plane of the, paper in the straight line OL, and the surface of the cone in the generating line OK; then OL : OK :: OL : OR, :: AN : AR, :: AX : AE, {Euclid, VI. 2.) I :: AX : AS, :: CA : OS, (Chap. III. Prop. II.) where C is the middle point of A A' and therefore the centre of the hyperbola. .'. KOL is half the angle between the asymptotes. (Chap. III. Prop. XVI.) Again, if EC be the conjugate semi-axis, and CU'U be drawn parallel to Er meeting OL)', OcV in ?7and U', then since CU : AG :: BL : OL, and CU' :A'C :: rL : OL, CU . CU' : AG 2 :: EL . rL : OL 2 , • XL' : OL 2 y but EC' 2 : AC 2 :: KL 2 : OL* ) . EC* = CU . CU. Coe. 2. If the cutting plane is parallel to the axis OL and OL coincide. 120 CONIC SECTIONS. In this case half the angle between the asymptotes of the hyperbolic section is equal to the constant angle DO I, and we can at once see that OC is the semi-conjugate axis. This affords a convenient method of obtaining a pair of conj agate hyperbolas. Draw Oi at right angles to 01 in the plane of the paper, and let another cone be formed by supposing OD to revolve round Oi in such a manner that the angle DOi is the same in all positions, and equal to the complement of DOI. Then if through any point A on the common generating line OD we draw two planes at right angles to the plane of the paper, and parallel respectively to OI and Oi, they will cut the cones in two hyperbolas, whose semi-transverse axes will be respectively AG, OC, and whose semi-conjugate axes' will be respectively OC, AC, and which therefore will be conjugate to each other. COXIC SECTIONS. 121 70. As long as the cutting plane remains parallel to itself, it is evident that the ratio of AE to AX, and therefore of AS to A X will be altered. Hence the sections made by planes inclined at the same angle to the axis of the cone will have the same eccentricity * 71. Through any point Q on the circle EQc let a plane be drawn parallel to ANP, intersecting the plane of the paper in the straight line WLN', the cone in the curve W Q P', and the planes of the circular sections EQc, BPr in the ordinates QL, P'N'. Then it is manifest that the curve WQF' will touch the circle wQw\ formed by the intersection of the cutting plane with the sphere at the extremities of the ordinate QL pro- duced. Join OP' meeting EQe in Q' ; then P'Q' : XL :: BE : XX, :: SA : AX. * The ratio of S A : AX, or of CS : CA, is called the eccentricity. 122 CONIC SECTIONS. ButP'Q' is equal to the tangent drawn from P r to the circle wQw', and N' L is equal to the perpendicular from P' on the common ordinate of the circle wQw' and the section WQP\ Hence we have the following important property, — If a circle touch a conic section in two points at the extremities of an ordinate, the ratio which the tangent drawn to the circle from any point on the curve bears to the perpen- dicular from the same point on the common chord is equal to the eccentricity of the conic section. If the two points in which the circle touches the conic coincide, the circle becomes the circle of curvature at the vertex ; and therefore the ratio which the tangent drawn from any point of a conic section to the circle of curvature at the vertex bears to the abscissa of the point, is constant and equal to the eccentricity of the curveo PEOBLEMS ON THE SECTIONS OF THE CONK 1. The foci of all parabolic sections which can be cut from a given right cone lie upon the surface of another cone. 2. The foci of all elliptical sections of a given right cone, in which the ratio of CA to CS is the same, will lie on two other cones. 3. The extremities of the minor axes of the elliptical sec- tions of a right cone made by parallel planes lie on two generating lines. 4. The latus rectum of a parabola cut from a given cone varies as the distance between the vertices of the cone and the parabola. 5. Under what conditions is it possible to cut an equilateral hyperbola from a given right cone ? 6. Two cones whose vertical angles are supplementary are joined as in Art. 69, Cor. 2. Prove that the latera recta of the curves of section of the cones, whose axes are respectively 01 and Oi, made by planes parallel or perpendicular to the plane of the axes, are in the duplicate ratio of Oi and 01. ADDITIONAL PEOBLEMS. 1. Show that the part of the directrix of a parabola, inter- cepted between the perpendiculars on it from the extremities of any focal chord, subtends a right angle at the focus. 2. The ]ocus of the foci of all parabolas touching the three sides of a triangle is a circle. Prove this, and give a geo- metrical construction for rinding the centre. 3. A system of parabolas which always touch two given straight lines have their axes parallel ; show that the locus of the foci is a straight line. 4. Prove that the locus at the foot of the perpendicular from the focus of a parabola on the normal is a parabola. 5. If S be the focus of a parabola, which touches the sides AB, AC of the triangle ABC at the points B, C, and the centre of the circle described about the triangle ; prove that the angle OS A is a right angle. G. Prom the focus of a parabola a straight line is drawn parallel to the tangent at any point B, meeting the diameter through B in V ; show that the tangent drawn from B to any circle passing through V is equal to one-half of the ordinate Q V, V being the second point in which the circle cuts the diameter through P. 7. B Sp is a focal chord, and upon PS and pS as diameters, circles are described ; prove that the length of either of their common tangents is a mean proportional between A 8 and Bp. CONIC SECTIONS. 125 8. If A Q be a chord of a parabola through the vertex A, and QR be drawn perpendicular to A Q to meet the axis in R ; prove that AR will be equal to the chord through the focus parallel to A Q. 9. The locus of the vertices of all parabolas, which have a common focus and a common tangent, is a circle. 10. Two parabolas have a common axis and vertex, and their concavities turned in opposite directions; the latus rectum of one is eight times that of the other ; prove that the portion of a tangent to the former, intercepted between the common tangent and axis, is bisected by the latter. 11. B is a point on a radius OA of a circle, whose centre is 0. On OA produced a point G is taken, such that OB . 00 = OA a . If P be any point on the circumference of this circle, R the middle point of BF, and Q the point of intersection of AR, CF ; prove that the locus of Q is a circle. 12. If from the middle point of a focal chord of a parabola two straight lines be drawn, one perpendicular to the chord meeting the axis in 67, and the other perpendicular to the axis meeting it in N\ show that NG is constant. 13. A circle is drawn touching the axis of a parabola, the focal distance of a point F, and the diameter through P. Show that the locus of its centre is a parabola with vertex S, and latus rectum equal to AS. 14. If from the point of intersection of the directrix and axis of a parabola a chord XFQ be drawn, cutting the parabola in F, Q ; show that the rectangle contained by the ordinates of FQ is equal to the square of one-half the latus rectum. 15. Find the locus of the centre of a circle which touches a given circle and a given straight line. 16. Given one point of contact of a parabola witli three 126 CONIC SECTIONS. tangents given in position, find the two other points of contact. 17. The triangle ABC circumscribes a parabola whose focus is S. Through A, B, G, lines are drawn perpendicular respectively to SA, SB, SC. Show that these lines pass through one point. 18. From the focus a line is drawn parallel to the tangent atP, meeting the parabola at Q. QNis an ordinate, and the tangents at P and Q meet the axis in T and T. Prove that SN* = 4: AT . AT\ and that if the diameter at P meet SQ in P, the locus of P is a parabola, whose latus rectum is half that of the given parabola. 19. P is any point in a parabola ; through S a line is drawn at right angles to the axis, meeting the chord AP or AP produced in P. Prove that SK . SB = 2 A S . S Y, where S Y is the perpendicular on the tangent, and SK on the normal. 20. Prom the focus S of a parabola SK is drawn, making a given angle with the tangent at P. Show that the locus of K is that tangent to the parabola which makes with the axis an angle equal to the given angle. 21. PSQ is a focal chord of a parabola, AP' a parallel chord meeting the latus rectum in Q' ; prove that AP' . AQ = SP . SQ. 22. The circle of curvature at any point of a parabola whose abscissa is A JY cuts the axis in U and U ', Prove that AU,AU' = 3AN 2 . 23. AB is a diameter of a circle. Prom any point Q in the circumference a tangent QP is drawn, and from P a per- pendicular PN is let fall upon AB. Show that if P be always taken so that QP is equal to AN, the locus of P will be a parabola. 24. It a tangent be drawn from any point of a parabola to the circle of curvature at the vertex, the length of the tangent CONIC SECTIONS. 127 will be equal to the abscissa of the point measured along the axis. 25. To two parabolas which have a common focus and axis two tangents are drawn at right angles ; the locus of their intersection is a straight line parallel to the directrices. 2G. If any three tangents be drawn to a parabola, the circle described about the triangle so formed will pass through the focus, and the perpendiculars from the angles on the opposite sides intersect in the directrix. 27. A parabola touches one side of a triangle in its middle point, and the other two sides produced. Prove that the per- pendiculars drawn from the angles of the triangle upon any tangent to the parabola are in harmonical progression. 28. Two equal parabolas have the same axis and vertex, but are turned in opposite directions ; chords of one parabola are tangents to the other. Show that the locus of the middle point of the chords is a parabola whose latus rectum is one- third of that of the given parabola. 29. Two equal parabolas have the same focus, and their axes are at right angles to each other, and a normal to one of them is perpendicular to a normal of the other ; prove that the locus of the intersection of such lines is a parabola. 30. Show that in every ellipse there are two equal conju- gate diameters, coinciding in direction with the diagonals of the rectangle, which touches the ellipse at the extremities of the axes. 31. If a circle be described through the two foci of an ellipse, cutting the ellipse, show that the angle between the tangents to this circle, and to the ellipse at either point of intersection, is equal to the inclination of the normal to the ellipse to the axis minor. 32. The points in which the tangents at the extremities of the transverse axis of an ellipse are cut by the tangent at any 128 CONIC SECTIONS. point of the curve, are joined one with each focus ; prove that the point of intersection of the joining lines lies in the normal at the point. 33. The external angle between any two tangents to an ellipse is equal to the semi-sum of the angles which the chord joining the points of contact subtends at the foci. 34. The tangent to an ellipse at any point P is cut by any two conjugate diameters in 1\ t, and the points T, t, are joined with the foci S, S' respectively; prove that the triangles SPT, S' Pt are similar to each other. 35. P is any point on a fixed circle, the centre of which is ; E is a fixed point without the circle ; an ellipse is described with centre and area constant so as always to touch EP at P. Find the locus of the extremities of the diameter conjugate to OP. 36. The normal at any point P of an ellipse cuts the axes in G, g ; prove that if any circle be described passing through G, g, the tangent to it from P is equal to CD. 37. Given a focus, a tangent, and the eccentricity of a conic section ; prove that the locus of the centre is a circle. 38. A straight line is drawn through a given point C within a circle to cut it in P, P'. If p is taken in it such that Cf = GP . CP', find the locus of p. 39. In the ellipse PY . PT : PA 72 :: CS 2 : PC 2 and ST. CD = SP. EC. 40. Show that if the distance between the foci of the ellipse be greater than the length of its axis minor, there will be four positions of the tangent, for which the area of the triangle, included between it and the straight lines drawn from the centre of the curve to the feet of the perpendiculars from the foci on the tangent, will be the greatest possible. CONIC SECTIONS. 129 41. Two conjugate diameters of an ellipse are cut by the tangent at any point P in M, N; prove that the area of the triangle GPM varies inversely as that of the triangle CPX. 42. Circles are described on SY, S'Y' as diameters, cutting SP, S'P respectively in Q, Q'. Trove that SQ . S'P = SP . S'Q' = BC 2 . 43. PSP', 2 1 ^v' are anv two focal chords of a conic section, P and p being on the same side of the axis ; prove that Pp, P ' p' meet on the directrix. 44. Prove that an ellipse can be inscribed in any parallelo- gram so as to touch the middle points of the four sides ; and show that this ellipse is the greatest of all inscribed ellipses. 45. If from any point on the exterior of two concentric, similar, and similarly placed ellipses, two tangents be drawn to the interior ellipse which also intersect the exterior; show- that the distance between the points of intersection will be double of that between the points of contact. 4G. The tangent at any point P in an ellipse, of which S and // are the foci, meets the axis major in T, and TQli bisects HP in Q, and meets SP in P ; prove that PR is one- fourth of the chord of curvature at P through S. 47. Trove that the distance between the two points on the circumference of an ellipse at which a given chord, not passing- through the centre, subtends the greatest and least angles, is equal to the diameter which bisects that chord. 48. From any point on the auxiliary circle chords are drawn through the foci of an ellipse, and straight lines join the extremities of the chords with the extremity of tin 1 diameter passing through the point; prove that these lines will touch the ellipse. 49. A quadrilateral circumscribes an ellipse. Trove that either pair of opposite sides subtends supplementary angles at either focus. K 130 CONIC SECTIONS. 50. Two tangents to an ellipse intersect at right angles ; show that the straight line joining their point of intersection with the point of intersection of the normals at the points of contact passes through the centre. 51. P, Q are points in two confocal ellipses, at which the line joining the common foci subtends equal angles ; prove that the tangents at P, Q are inclined to an angle which is equal to the angle subtended by PQ at either focus. 52. Tangents to an ellipse are drawn from any point in a circle through the foci ; prove that the lines bisecting the angle between the tangents all pass through a fixed point. 53. If the ordinate at P meet the auxiliary circle in Q, and CQ meet the ellipse in II, then G R is equal to the per- pendicular on the tangent at P from C. 54. If P be a point such that SP, >S" P are perpendicular; prove that CD 1 = 2 . BO 2 . 55. If circles be described to the triangle SPS' opposite to the angks S and S' ; prove that the rectangle contained by their radii is equal to PC 2 . 56. The circle of curvature at any point P of an ellipse meets the focal distances in B, B' ; SU is a tangent to the circle. Prove that SU* : SP 2 :: 2 . JSP - 3 . AG : AG, and if EH' passes through the centre of the circle of cur- vature, CP = OS. Determine the limits of possibility in both cases. 57. A straight line is drawn from the centre of an ellipse meeting the ellipse in P, the circle on the major axis in Q, and the tangent at the vertex in T. Prove that as G T ap- proaches and ultimately coincides with the semi-major axis, TP and Q T are ultimately in the duplicate ratio of the axes. CONIC SECTIONS. lol 58. A straight line is drawn through the focus S of an ellipse meeting the two tangents at right angles to it in Y and Z, the diameter parallel to these tangents in L, and the directrix in M; prove that SL : SY :: SZ : SK 59. If any equilateral triangle PQJl he described in the auxiliary circle of an ellipse, and the ordinates to P. Q, R meet the ellipse in P\ Q', Ii' ; the circles of curvature at P', Q', R\ meet in one point lying on the ellipse. GO. From a point T two tangents TP, TQ are drawn to an ellipse. Show that a circle with T as centre can be described so as to touch SP, S'P, SQ, S'Q. 61. If the normal at P meet the axis minor in j, and if the tangent at P meet the tangent at the vertex A in V; show that Sg : SC :: PV : VA. 62. If a circle passing through Y and Y touch the major axis in Q y and that diameter of the circle which passes through Q meet the tangent in P; show that PP = BO. (See fig. Prop. XV.) 63. If PG the normal at P cut the major axis in c7, and if DP, PN be the ordinates of D and P, prove that the tri- angles PGN, PPG are similar; and thence deduce that PG bears a constant ratio to CD. 64 The tangent at a point Pof an ellipse meets the tangents at the vertices in V, V. On VV as diameter, a circle is described which intersects the ellipse in Q, Q' ; show that the ordinate of Q is to the ordinate of P as B G to BG + GD where GD is conjugate to OP. 65. POP' is any diameter of an ellipse; the tangents at any two points Pand E' intersect in P; PE\ P'E intersect in G. Show that FG is parallel to the diameter conjugate to PGP'. K 2 132 CONIC SECTIONS. 66. If Pbe any point on an ellipse, and with P as centre and the semi-axis minor as radius a circle be described; prove that if PG be the normal, a circle described on OG as diameter will cut the first circle at right angles. 67. ABC is an isosceles triangle having AB = AC. BD, BE drawn on opposite sides of B C\ and equally inclined to it, meet AG in D and E. If an ellipse be described about BDE having its minor axis parallel to B C ; then AB will be a tangent to the ellipse. 6S. If AQ be drawn from one of the vertices of an ellipse perpendicular to the tangent at any point P; prove that the locus of the point of intersection of PS and QA produced will be a circle. 69. If Y, Y' be the feet of the perpendiculars from the foci of an ellipse on the tangent at P ; prove that the circle cir- cumscribed about the triangle YN Y' will pass through C. 70. Prove that the angle between the tangents to the auxiliary circle at Y t Y' is the supplement of the angle SPS'. 71. Pis any point on an ellipse ; PJIJ] PiV perpendicular to the axes meet respectively, when produced, the circles described on the axes as diameter in the points Q, Q\ Show that QQ' passes through the centre. 72. Assuming that the greatest triangle which can be in- scribed in a circle is equilateral, prove by the method of projection, that the greatest triangle which can be inscribed in an ellipse has one of its sides bisected by a diameter of the ellipse, and the others cut in points of bisection by the conjugate diameter. CONIC SECTIONS. 133 73. PQ is a chord of an ellipse, normal at P, LCL' the diameter bisecting it. Show that PQ bisects the angle LPL' and that LP -f L' P is constant. 74. A tangent to an ellipse at a point P intersects a fixed tangent in T; if through the focus a line be drawn perpen- dicular to ST meeting the tangent to P in Q ; show that the locus of Q is a straight line touching the ellipse. 75. In an ellipse if a line be drawn through the focus making a constant angle with the tangent ; prove that the locus of the point of intersection with the tangent is a circle. 76. Any chord PP' of an ellipse is produced to a point Q, such that P' Q is equal to half the diameter parallel to PP', and QR'R is drawn through the centre to meet the ellipse in A', R' ; show that the area PCP is three times the area P'CPi. 77. In an ellipse, L is the extremity of the latus rectum, and CD conjugate to CL. If a circle be described with centre C and passing through B, and a line be drawn through D parallel to the major axis, the portion of this line which lies within the circle will be equal to the latus rectum. 78. If P be any point in an ellipse, and K the point in which a normal at P intersects a line at right angles to it through S' t E the point of intersection of SP, and the diameter conjugate to OP, and if EK and GK be joined, each of the figures SOKE, S'GEK will be a parallelogram. 70. If T be a point on the axis A A ' produced, and PN the ordinate of the point where the tangent from T touches the ellipse ; prove that AN . A'JST : AT . A'T :: CN : CT. 80. Given in an ellipse a focus and two tangents ; prove that the locus of the other focus is a straight line. 134 CONIC SECTIONS. 81. A focus, a tangent, and the axis major being given, prove that the locus of the other focus is a circle. 82. A focus, a tangent, and the axis minor being given, prove that the locus of the other focus is a straight line. 83. An ellipse touches a fixed ellipse and has a common focus with it ; if the major axis be fixed, the locus of the other focus is a circle ; if the minor axis be fixed, the locus is an ellipse. 84. An ellipse and a parabola have a common focus. Prove that the ellipse either intersects the parabola in two points, and has two common tangents with it, or else does not cut it. 85. If in the ellipse a focus, a point, and the axis minor be given, the locus of the other focus is a parabola. 86. If at the extremities P, Q of any two diameters CP, CQ of an ellipse, two tangents Pp, Qp be drawn cutting each other in T, and the diameter produced in^> and q, then the area of the triangles TQp, TPq are equal. 87. If a straight line CN be drawn from the centre to bisect that chord of the circle of curvature at any point P of an ellipse, which is common to the ellipse and circle, and if it be produced to cut the ellipse in Q, and the tangent in T ; prove that CP = CQ, and that each is a mean proportional between OAand CT. 88. An ellipse is described so as to touch the three sides of a triangle ; prove that if one of its foci move along the cir- cumference of a circle passing through two of the angular points of the triangle, the other will move along the circum- ference of another circle, passing through the same two angular points. Prove also that if one of these circles pass through the centre of the circle inscribed in the triangle, the two circles will coincide. 89. A triangle is described about an ellipse, so that the extremities of one of its sides lie in an ellipse, confocal with the given one; prove that the line bisecting the opposite angle passes through the pole of that side with respect to the outer ellipse. CONIC SECTIONS. 135 00. Prove the following construction for a pair of tangents from any external point T to an ellipse of which the centre is C. Join CI 1 ; let TPCP' T, a similar and similarly situated ellipse, be drawn, of which C T is a diameter, and P, P' its points of intersection, with the given ellipse ; TP, TP' will be tangents to the given ellipse. 91. The locus of the foci of all ellipses inscribed in the same parallelogram is a rectangular hyperbola. Prove this, and give a geometrical construction for finding the asymp- totes. 92. AC is a fixed diameter of a circle, A BCD a quadri- lateral figure inscribed in the circle ; prove that if the angles BA C, J) AC be complementary, the locus of the intersection of BA, CD will be an hyperbola. 93. Trove that a circle can be described so as to touch the four straight lines drawn from the foci of an hyperbola to any two points on the same branch of the curve. 94. Any three diameters of an ellipse, LL\ MM', XX', being taken, a circumscribing parallelogram R T U V touches the ellipse at L, L',j\T, M\ Show that a conic sectioncan.be described through the points R, 1\ U, V, X y X', which will be an hyperbola whose asymptotes are the lines forming in the ellipse the diameters conjugate to XX' and to the other common chord of the ellipse and hyperbola. 95. On opposite angles of any chord of a rectangular hyperbola are described equal segments of circles ; show that the four points in which the circles to which these segments belong again meet the hyperbola, are the angular points of a parallelogram. 90. A triangle is inscribed in a rectangular hyperbola : prove that the circle described through the middle points of the sides of the triangle passes through the centre of the hyperbola. 136 CONIC SECTIONS. 97. ACB is an isosceles triangle; AB the base, and D any point in GB or GB produced : if BZ be drawn parallel to AD, meeting CA or GA produced in Z, prove that the middle point of DZ will be in an hyperbola whose asymp- totes are GA, GB. 98. An ellipse and hyperbola are described so that the foci of each are at the extremities of the transverse axis of the other ; prove that the tangents at their points of intersection meet the conjugate axis in points equidistant from the centre. 99. In a rectangular hyperbola, PK, PL are drawn at right angles to A'P, AP respectively to meet the transverse axis in K and L ; prove that PK is equal to AP and KL to uAA' , and the normal at P bisects KL. 100. In a rectangular hyperbola PC is a fixed diameter, Q any point on the curve ; show that the angles QPG, QGP differ by a constant angle. 101. If the tangent at any point Pof an hyperbola cut an asymptote in T, and if JSP cut the same asymptote in Q, then 8Q = QT. 102. If a given point be the focus of any hyperbola, passing through a given point and touching a given straight line, prove that the locus of the other focus is an arc of a fixed hyperbola. 103. At any P of an hyperbola a tangent is drawn, and PQ is taken on it in a constant ratio to GD ; prove that the locus of Q is an hyperbola. 104. In an hyperbola, supposing the two asymptotes and one point of the curve be given in position, show how to construct the curve ; and find the position of the foci. 105. If A, D be two fixed points, and the angle PAD always exceed PDA by a given angle ; find the locus of P, and the position of the transverse axis and asymptote. CONIC SECTIONS. 137 106. From the middle point D of the base AB of the triangle ABC a straight line EBB' is drawn, making a cjiven angle with A B, and the points B, B' are taken so that BB = B'B = I AB. If CA, CB take all possible positions consistent with the condition that the difference of the angles CAB, CBA is eqnal to EDA; prove that the point C will trace out a rectangular hyperbola of which AB, B'B are conjugate diameters. 107. In the rectangular hyperbola, prove that the triangle, formed by the tangent at any point and its intercepts on the axes, is similar to the triangle formed by the straight line joining that point with the centre, and the abscissa and semi-ordinate of the point. 108. Tangents are drawn to an hyperbola, and the portion of each tangent intercepted by the asymptotes is divided in a constant ratio ; prove that the locus of the point of section is an hyperbola. 109. Show that the point of trisection of a series of con- terminous circular arcs lie on branches of two hyperbolas, and determine the distance between their centres. 110. Prom a point B on one asymptote BB is drawn touching the hyperbola in B, and E T, BV are drawn through B, parallel to the asymptotes, cutting a diameter in T and V; BVis joined, cutting the hyperbola in P, p : show that TB, Tp touch the hyperbola. 111. Given in the ellipse a focus and two points, the locus of the uther focus is an hyperbola. 112. If a rectangular hyperbola passes through three given points, the locus of its centre is a circle, which passes through the middle points of the lines joining the three given points. 113. If the tangent at Pmeet one asymptote in T t and a line TQ be drawn parallel to the other asymptote to meet the curve in Q ; prove that if BQ be joined and produced both ways to meet the asymptotes in B and B\ BB' will be trisected at the points B and Q. 138 CONIC SECTIONS. 114. If two concentric rectangular hyperbolas have a com- mon tangent, the lines joining their points of intersection to their respective points of contact with the common tangent will subtend equal angles at their common centres. 115. If TP, TQ be two tangents drawn from any point T to touch a conic in P and Q, and if S and S' be the foci, then ST* : S'T 2 :: SP . SQ : ST . S'Q. 116. The circle of curvature at the vertex of a conic meets the axis again in D, and a tangent is drawn to the circle at D : if two tangents be drawn to the circle from any point in the conic they will intercept between them a constant length of the former tangent. 117. If the lines which bisect the angles between pairs of tangents to an ellipse be parallel to a fixed straight line, prove that the locus of the points of intersection of the tangents will be a rectangular hyperbola. 118. An hyperbola, of given eccentricity, always passes through two given points; if one of its asymptotes always pass through a third given point in the same straight line with these, prove that the locus of the centre of the hyperbola will be a circle. 119. A, P and B, Q are points taken respectively in two parallel straight lines, A and B being fixed, and P, Q variable. Prove that if the rectangle APBQ be constant, the line PQ will always touch a fixed ellipse or a fixed hyperbola, ac- cording as P and Q are on the same or opposite sides of AB. 120. If two plane sections of a right cone be taken, having the same directrix, the foci corresponding to that directrix lie on a straight line which passes through the vertex. 121. Give a geometrical construction by which a cone may be cut so that the section may be an ellipse of given eccen- tricity. CONIC SECTIONS. 139 122. Given a right cone and a point within it, there are but two sections which have this point for focus ; and the planes of these sections make equal angles with the straight line joining the given point and the vertex of the cone. 123. If the curve formed by the intersection of any plane with a cone be projected upon a plane perpendicular to the axis, prove that the curve of projection will be a conic section having its focus at the point m which the axis meets the plane of projection. 124. If F be the point where the major axis of an elliptic section meets the axis of the cone, and be the centre of the section ; prove that CF : C'S :: AA' : AO + A'O, being the vertex of the cone. SECOND SEPJES. 1. If OQ, OQ' be tangents to a parabola, and OF" drawn parallel to the axis meet the directrix in K and QQ' in l r , and QQ' meet the axis in X, OKXS shall be a parallelo- gram. 2. G is the foot of a normal at a point P of a parabola, Q is the middle point of SG, and X is the foot of the directrix, prove that QX 2 - QP- 2 = 4AS\ 3. Through any point tangents are drawn to a parabola ; and through straight lines are drawn parallel to the normals at the points of contact, prove that one diagonal of the parallelogram so formed passes through the focus. 4. Two parabolas have their foci coincident, prove that the common chord passes through the intersection of the direc- trices, and that they cut one another at angles which are half the angles between the axes. 140 CONIC SECTIONS. 5. A circle is drawn through the point of intersection of two given straight lines and through another given point, prove that the straight line joining the points, where the circle again meets the two given straight lines, touches a fixed parabola. G. Common tangents are drawn to two parabolas which have a common directrix and intersect in P, Q ; prove that the chords joining the points of contact in each parabola are parallel to PQ ; and the part of each tangent between its points of contact with the two curves is bisected by PQ produced. 7. The tangents at two points Q, Q' in the parabola meet the tangent at P in P, P' respectively, and the diameter through their point of intersection T meets it in K\ prove that PR = KB' and that if QM, Q'M/ TN be the ordinates of Q, Q,' T respectively to the diameter through P, PNis a mean proportional between PM and PM' . 8. P and p are points on a parabola on the same side of the axis and PN, pn the ordinates; the normals at P and p intersect in Q ; prove that the distance of Q from the axis _ 2 P N. pn{PN + p n) (latus rectum) 2 Deduce the value of the radius of curvature. 9. Two equal parabolas have a common focus, and from any point on the common tangent another tangent is drawn to each ; prove that these tangents are equidistant from the common focus. 10. From an external point two tangents are drawn to a parabola, and from the points where they meet the directrix two other tangents are drawn meeting the tangents from at A and B. Prove that AB passes through the focus S, and that OS is at right angles to AB. CONIC SECTIONS. 141 11. Given two tangent lines to a parabola and the focus, show how to determine the eurve. 12. Inscribe in a given parabola a triangle having its sides parallel to those of a given triangle. 13. Normals at P, p the extremities of a focal chord of a parabola meet the axis in G and g. The tangents meet in T. Prove that the circles about the triangles SPG, Spg intersect on T 8 produced at a point P such that S bisects P T. 14. A circle and parabola touch one another at both ends of a double ordinate to the parabola, prove that the latus rectum is a third proportional to the parts into which the abscissa of the points of contact is divided by the circle either internally or externally. 15. 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; then the rectangle contained by PP' and P' Pi is constant. 16. OP, OQ are two tangents to a parabola, and OM is drawn perpendicular to the axis ; if R be the point where PQ cuts the axis, MP is bisected by the vertex. 17. A parabola, whose focus is 8, touches the three sides of a triangle ABC, bisecting the base EC in D ; prove that A 8 is a fourth proportional to A D, A B, and A 0. 18. If the chord of contact be normal at one end, the tangent at the other is bisected by the perpendicular through the focus to the line joining the focus to the external point. 19. Two parabolas have a common focus, and from any point on their common tangent are drawn two other tangents to the parabolas; prove that the angle between them is equal to the angle between the axes of the parabolas. 20. If tangents be let fall on any tangent to a parabola from two given points on the axis equidistant from the focus, the difference of their squares is constant. 142 ccmc SECTIONS. 21. If a quadrilateral be inscribed in a circle, one of the three diagonals of the quadrilateral passes through the focus of the parabola which touches its four sides. 22. A chord of a parabola is drawn parallel to a given straight line, and on this chord as diameter a circle is de- scribed ; prove that the distance between the middle point of this chord, and of the chord joining the other two points of intersection of the circle and parabola will be of constant lenoth. 23. Through any point P of an ellipse a line is drawn perpendicular to the radius vector CP meeting the auxiliary circle in R, R' : prove that RE' is equal to the difference of the focal distances of the extremity of the diameter conjugate to CP. 24. An ellipse and parabola whose axes are parallel, have the same curvature at a point P and cut one another in Q ; if the tangent at P meet the axis of the parabola in T prove that PQ is four times PT. 25. A triangle is inscribed in an ellipse so that each side is parallel to the tangent at the opposite angle : prove that the sum of the squares on the sides is to the sum of the squares on the axes as nine to eight. 26. In an ellipse the perimeter of the quadrilateral formed by the tangent, the perpendiculars from the foci and the trans- verse axis, will be the greatest possible, when the focal dis- tances of the point of contact are at right angles to each other. 27. 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. 23. Perpendiculars SY, S r Y' are drawn from the foci upon a pair of tangents TY, TY' ; prove that the angles STY, S J TY' are equal or supplementary to the angles at the base of the triangle formed by joining F, Y' to the centre of the ellipse. CONIC SECTIONS. 143 29. With the centre of perpendiculars of a triangle as centre, are described two ellipses, one about the triangle, and the other touching its sides : prove that these ellipses are similar and their homologous axes at right angles. 30. Through a point P of an ellipse a line PDE is drawn cutting the axes so that the segments PI) and PE are equal to the two semi-axes respectively ; perpendiculars to the axes through D and E intersect in ; prove that PO is a normal. 31. If the focal distance SP of an ellipse meet the con- jugate diameter in E ; then the difference of the squares on UP and SE will be constant. 32. The chords of curvature through any two points of an ellipse in the direction of the line joining them are in the same ratio as the squares of the diameters parallel to the tangents at the points. 33. From the extremities of the diameter of an ellipse, perpendicular to one of the equi-conjugate diameters, chords are drawn parallel to the other. Prove that these chords are normal to the ellipse. 34. An ellipse of given semi-axes touches three sides of a given rectangle ; find its centre and foci. 35. If P, Q be any two points on a fixed ellipse, whose foci are S, //and if JSI\ II Q intersect within the ellipse at R, prove that two ellipses can be drawn touching each other at It, the one having 8 for focus and touching the given ellipse at Q, the other having II for focus and touching the given ellipse at P. If the major axis of one of the variable ellipses be given find the loci of their other foci, and of their point of contact. 36. Find the positions of the foci and directrices of an ellipse, which touches at two given points P, Q, two given straight lines PO, QO, and has one focus on the line PQ, the angle PO Q being less than a right angle. 144 CONIC SECTIONS. 37. If Pp be drawn parallel to the transverse axis of a given ellipse, meeting the ellipse in P,p, and the circle whose diameter is the conjugate axis in B, r, then shall Pp : Br :: QQ f : PP'. 38. Through any point P of an ellipse are drawn straight lines APQ; A' P R meeting the auxiliary circle in Q, B, and ordinates Qq, Br are drawn to the transverse axis; prove that, L being an extremity of the latus rectum Aq . A'r : At . A'q :: AC 2 : SL 2 . 39. Two ellipses whose axes are equal, each to each, are placed in the same plane with their centres coincident, and axes inclined to each other. Draw their common tangents. 40. If S, S' be the two foci of an ellipse ; and $ Y the perpendicular from S upon any tangent, prove that /S' Y will bisect the corresponding normal. 41. OB is a diagonal of the parallelogram of which OQ, OQ', tangents to an ellipse, are adjacent sides : prove that if R be in the ellipse, will lie on a similar and similarly situated concentric ellipse. 42. A circle passes through a focus of an ellipse, has its centre on the major axis of the ellipse, and touches the ellipse : shew that the straight line from the focus to the point of contact is equal to the latus rectum. 43. If the focus of an ellipse be joined with the point where the tangent at the nearer vertex intersects any other tangent, and perpendiculars be drawn from the other focus on the joining line, and the last mentioned tangent, prove that the distance between the feet of these perpendiculars is equal to the distance from either focus to the remoter vertex. 44. A parallelogram is described about an ellipse ; if two of its angular points lie on the directrices, the other two will lie on the auxiliary circle. CONIC SECTIONS. 145 45. PS is the focal chord of a point on an ellipse ; CR is a radius of the auxiliary circle parallel to PS, and drawn in the direction from P to S] SQ is a perpendicular on OR; shew that the rectangle contained by SP and QR is equal to the square on half the minor axis. 46. From the extremity P of the diameter PQ of an ellipse the tangent TPT is drawn meeting two conjugate diameters in T, T. From P Q the lines PR, QR are drawn parallel to the same conjugate diameters. Prove that the rectangle under the semi-axes of the ellipse is a mean proportional between the triangles P QR and CTT. 47. If CP and CD be equal conjugate diameters of an ellipse, and the tangent and normal at P meet the major axis in T and G respectively, prove that TC . TG = 20 P\ 48. An ellipse is inscribed in a triangle having its centre at the centre of the circumscribed circle of the triangle ; prove that the perpendiculars from the corners of the triangle on the opposite sides will be normals to the ellipse. 49. Tangents are drawn from any point on the auxiliary circle to the ellipse ; prove that the line joining one of the points of contact with one of the foci is parallel to the line joining the other point of contact with the other focus. 50. From a point P without an ellipse, P Q is drawn parallel to the major axis, Q being either of the points in which it meets the curve ; then the straight line bisecting PQ at right angles, the tangent at Q, and the line which joins the middle points of PK, PL (the tangents drawn from P) meet in a point. 51. CP and CD are conjugate semi-diameters of an ellipse ; PQ is a chord parallel to one of the axes ; shew that DQ is parallel to one of the straight lines which join the ends of the axes. 52. A series of ellipses pass through the same point and have a common focus and their major axes of the same length ; prove that the locus of their centres is a circle. What are the limits of the eccentricities of the ellipses ? L 146 CONIC SECTIONS. 53. A parallelogram is inscribed in an ellipse, and from any point on the ellipse two straight lines are drawn parallel to the sides of the parallelogram ; prove that the rectangles under the segments of these straight lines, made by the sides of the parallelogram, will be to one another in a constant ratio. 54 If the tangent and normal in an ellipse meet the axis in T and G respectively, and Q be the corresponding point of the auxiliary circle, then TQ : TP :: BC : PG. 55. POP' is a diameter of an ellipse, CD conjugate to CP; prove that PD, DP' are inversely proportional to the diameters which bisect them. 56. If PJFhe one side of a parallelogram described about an ellipse, having its sides parallel to conjugate diameters, and the lines joining E, F to the foci intersect in 0, 0', shew that 0, 0' and the foci will lie on a circle. 57. If the normal at any point of a hyperbola meet the conjugate axis in g, and S be the focus, Pg will be to Sg in a constant ratio. 58. In the rectangular hyperbola, if circles be described passing through the centre of the hyperbola and the points of intersection of the tangent with the pairs of conjugate diameters, their centres will lie on a fixed straight line. 59. A chord of a rectangular hyperbola is drawn perpen- dicular to a fixed diameter and the extremities of the chord and diameter are joined by four straight lines ; then the line joining the two fresh points of intersection of these lines will be parallel to a fixed line. 60. A series of confocal ellipses is cut by a confocal hyper- bola ; prove that either focal distance of any point of inter- section is cut by its conjugate diameter with respect to the particular ellipse in a point which lies on a circle. CONIC SECTIONS. 147 61. If from any point on the hyperbola a tangent be drawn to the circle described on the transverse axis as diameter, its length is equal to the semi-minor axis of the confocal ellipse through the point. 62. 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 will be similar. 63. Two tangents are drawn to the same branch of a rectangular hyperbola from an external point ; prove that the angles which these tangents subtend at the centre are respectively equal to the angles which they make with the chord of contact. 64. A circle is described through P, P' , the extremities of any diameter of a rectangular hyperbola, and cutting the tangent at P and T: prove that P'T and the tangent to the circle at P meet on the hyperbola. Go. A fixed hyperbola is touched by a concentric ellipse. If the curvatures at the point of contact are equal, the area of the ellipse will be constant. 66. Two equal circles touch a rectangular hyperbola at a point 0, and intersect it again in P, Q ; P\ Q' respectively; prove that the points may be so taken that PP', QQ' each subtend a right angle at 0, and that the straight lines joining P', Q' to the centre of the circle OPQ will trisect OP, OQ respectively. 67. In a central conic given a focus, the length of the transverse axis, and that the second focus lies on a fixed straight line ; prove that the conic will touch two fixed parabolas having the given focus for focus. 68. In a hyperbola a circle is described about a focus as centre, with radius one-fourth of the latus rectum ; prove that the focal distances of the points of intersection with the hyperbola are parallel to the asymptotes. L 2 148 CONIC SECTIONS. G9. Prove that the angles subtended at the vertices of a rectangular hyperbola by any chord parallel to the conjugate axis are supplementary. 70. A line moves in such a manner that the sum of the spaces on its distances from two given points is constant ; prove that it always touches an ellipse or hyperbola, the square on whose transverse axis is equal to twice the sum of the squares on the distances of the moving line from the given points. 71. The eccentricity of a hyperbola is 2. A point D is taken on the axis, so that its distance from the focus 8 is equal to the distance of S from the further vertex A', Pbein^ any point on the curve, A'P meets the latus rectum in K. Prove that DK and SP intersect on a certain fixed circle. 72. In a hyperbola if PIT, PK be drawn parallel to the asymptotes, and a line through the centre meet PH, PK in R, T, and the parallelogram PRQT be completed, Q is a point on the hyperbola. 73. If an ellipse and hyperbola have their axes coincident and proportional, points on them equidistant from one axis have the sum of the squares on their distances from the other axis constant. 74. In the hyperbola prove that (see fig. Prop. XXIII.) MD : PN :: AG : EG :: GN : CM. 75. A parabola and hyperbola have the same focus and directrix, and SPQ is a line through the focus S to meet the parabola in P, and the nearer branch of the hyperbola in Q ; prove that PQ varies as the rectangle contained by 8P and 8Q. 7G. If two hyperbolas have their transverse axes parallel, and their eccentricities equal, they will have parallel asymp- totes. Does the converse hold ? 77. PP r is any diameter of a rectangular hyperbola, Q 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 QR' are equal. CONIC SECTIONS. 149 i '$. In a hyperbola a line parallel to BC, through the inter- section of the tangent at P with the asymptote, meets DP, CA produced in the same point, CD and CP being con- jugate. 70. From a given point in a hyperbola draw a straight line such that the segment intercepted between the other intersection with the hyperbola and a given asymptote shall be equal to a given line. When does the problem become impossible 1 80. A tangent is drawn from any point on the transverse axis of a hyperbola to the auxiliary circle ; if the anglu between this tangent and the ordinate be called the eccentric angle, shew that the eccentric angles at the points of inter- section of an ellipse with two hyperbolas, confocal with itself, are equal. 81. Two unequal parabolas have a common focus and axes opposite ; a rectangular hyperbola is described touching both parabolas and having its centre at the common focus ; prove that the angle between the lines joining the points of contact to the common focus is 60°. 82. The tangent and normal at any point of a hyperbola intersect the asymptotes and axes respectively in four points which lie on a circle passing through the centre of the hyper- bola ; and the radius of this circle varies inversely as the perpendicular from the centre upon the tangent. 83. Prove that the rectangular hyperbola which has for its foci the foci of an ellipse, will cut the ellipse at the extremities of the equal conjugate diameters. 84. ABC is a given triangle. CA, CB produced are the asymptotes of a hyperbola cutting AB in P. Find the position of P when the sum of the squares on its axes is the greatest possible. 85. From the point of intersection of an asymptote and a directrix of a hyperbola a tangent is drawn to the curve ; prove that the line joining the point of contact with the focus is parallel to the asymptote. 150 CONIC SECTIONS. 86. Find, the position and magnitude of the axes of a hyperbola which has a given line for asymptote, passes through a given point, and touches a given straight line in a given point. 87. Prove that a hyperbola can be described passing through the extremities of any two diameters of a given ellipse, having diameters conjugate to these for its asymp- totes. 88. P Q is a normal at a point P of a rectangular hyperbola meeting the curve again in Q ; prove that P Q is equal to the diameter of curvature at P. 89. P is a point on a hyperbola whose foci are S and //; another hyperbola is described whose foci are S and P, and whose transverse axis is equal to SP — 2PII; shew that the hyperbolas w 7 ill meet only at one point, and that they will have the same tangent at that point. 90. The tangent at any point on a hyperbola is produced to meet the asymptotes, thus forming a triangle ; determine the locus of the point of intersection of the straight lines drawn from the angles of this triangle to bisect the opposite sides. 91. The extremities of the latera recta of all conies which have a common major axis lie on two parabolas. 92. Having given three points, prove that there are four straight lines such, that with any one of them as directrix, and. any one of the given points as focus, a conic section may be described passing through the other two given points. 93. In any conic section if P G, pg, the normals at the ends of a focal chord intersect in 0, the straight line through parallel to Pp bisects G and 4. Prop. III. If 4i? is harmonically" divided in and 0', and AB be bisected in (7; then 00 . CO' = CB' 2 a u a jj o' Since AO : OB :: 40' : 0'i?, .-. 40 + 0£ : 40 - 01? :: 40' + O'B : AO' - O'B- or 2GB : 2C0 :: 2 CO' : 2CB, .-. CO . CO' = 0B\ Dek When the straight line A B and therefore also C is given points and 0' chosen so that CO . CO' = CB 2 are said to be in involution. Such points therefore divide AB harmonically. 73. Def. If a straight line AB be divided in any two points and 0', either both internal or both external, or one internal and the other external, and without any limitation as to the side of 4 B on which the external point or points should fall, the ratio of the ratios AO : OB and AO' : O'B, or, as we shall express it, the ratio (AO : OB) : (AC : O'B) is called the «?iharmonic ratio of the range AOBO', which is always thus represented by AOBO', whatever maybe the geometrical order in which the letters and 0' appear with reference to 4 and B. The imkarmonic ratio of the range AO' BO is the ratio {AO' : O'B) : (AO : OB), and is therefore the reciprocal of the a?iharnionic ratio of the range AOBO'. 156 APPENDIX. If tlie points and 0' are both internal or both external with regard to A and B, then the (inharmonic ratio of the range A 0B0\ viewed algebraically, is positive; when one point of division is external and the other internal it is negative. "When therefore the straight line AB is harmonic- ally divided in and 0', the : OB) : (ao : ob) :: (A 0' : O'B) : (ao' : o'h), or (.4 : 0£) : (40' : O'B) :: (ao : oh) : (ao' : o'6). Prop. V. The straight lines joining the intersections of the diagonals of a quadrilateral figure with the points of intersection of a pair of opposite sides are divided harmonically at the points where they meet the other two sides ; also the sides of the quadrilateral are divided harmonically by these straight lines. Let ABOD be a quadrilateral, and let the diagonals AC, BE, intersect in E, and the pairs of opposite sides AD, BC, and AB, D C, in F and G respectively. Let EF meet the sides CE, AB, in P, Q ; and EG the sides A D, B C, in B, S. Since the four straight lines EA, EQ, EB, EG, forming the pencil E(AQBG)sive intersected by the straight line DBCG in the points C, F, D, G, the ranges AQBG^CPDG, have the same anharmonic ratio. Again, since the four straight lines FA, FQ, FB, FG, forming: the pencil F (AQBG) are intersected by the straight line DPCG in the points D, P, C, G, the ranges A QBG, DPCG, have the same anharmonic ratio. 158 APPENDIX. . \ the ranges CPDG, DPCG, have the same anharmo-nic ratio. But the anharmonic ratio of the range CPDG, viz. the ratio (OP : PD) {CG : GD\ is the reciprocal of the anharmonic ratio of the range, DPGG, viz. the ratio {DP: PC) : (D G : GG), .'. the ranges DPCG and CPD G are harmonic {Art. 73). Hence also the ranges BESG, AQBG, are harmonic ; and exactly in the same way it may be proved that the ranges PCSB, FPEQ, FDR A, are harmonic. Con. If A C, BD, he produced to meet FG in II and K respectively; then AC, BD, are divided harmonically in E, II, and E, K respectively ; and GF is divided harmonically in //, K. IST.B. — The point K will be on FG or GF produced, ac- cording as GH is less or greater than HF. If BE be equal to ED the point K is at infinity, and GF, BD are parallel, and GFis also bisected in H. Pkop. VI. 75. If from an external point a pair of tangents OP, OP be drawn to any conic, and a straight line OQQ' intersect the APPENDIX. 159 curve in ft Q' and PP' in 0' ; then Q Q' is divided harmo- nically in 0' and 0. Through draw V bisecting PP' in V, and draw the double ordinates RQvq, R'Q'v'q parallel to PP meeting OV in v and v' ; then Now Qq and ft^' are bisected in v and v' i?P 2 : R'P 2 .-.RP : R'P .-. GO': O'ft RQ . Bq : R'Q' . .SV Rv*-Qv*: 72 V 2 - Q'v' 2 Ov 2 : Ov' 2 OR 2 : OB' 2 OR : 0i2' 00 : OQ' r. QQ' is divided harmonically in 0, 0' ; and therefore also (see Prop. II.) 0' is divided harmonically in Q and Q'. 160 APPENDIX. Cob. If the conic is a parabola, OV is drawn parallel to the axis. If an ellipse or hyperbola, V is drawn through the centre. If the point be the centre of the hyperbola, then, OP, OP 7 are asymptotes, and the line PP' is at infinity, and any chord Q Q' through G is bisected at C, while the fourth point which with harmonically divides QQ' is at an in- finite distance. Prop. VII. 76. The locns of the point of intersection of the tangents at the extremities of any chord drawn through a given point either within or without a conic is a straight line. Let be the given point. Through draw any chord QOQ', and bisect QQ' in V. First let the curve be a parabola. Through and V draw 0P0' and VWK parallel to the axis of the parabola meeting the curve in P and W. Make WK equal to V W and PO' equal to PO APPENDIX. 161 Then K is the point of intersection of the tangents at Q and Q\ From IF draw the tangent W U parallel to QQ' and the ordinate WR parallel to the tangent at P meeting r in T] and R respectively. Join KO'. Now PO' — PO by construction and PR = PU .-. O'R = OU = wv .-. KO' is parallel to IV R, and therefore to the tangent at P. .•. the locus of K is the straight line drawn through 0' parallel to the tangent at P. Next let the curve be a central conic whose centre is C. 162 APPENDIX. Through C draw COPO\ CVWK meeting the conic in Pand W. Take CO' , CK third proportionals respectively to CO, CP, and CV, OW, so that CO. C0'= 6 7 P 2 and CV . CK = CW\ Then K is the point of intersection of the tangents at Q and Q'. Through W draw the tangent IV U parallel to QQ\ and the ordinate WR parallel to the tangent at P meeting CP in U and R respectively. T oin KO'. Now CR. CU = CR : CO' CP 2 CO . CO CV CIV CO' CU : CW : CK by construction. .'. KO' is parallel to WR, and therefore to the tangent at P. .'. the locus of K is the straight line drawn through 0' parallel to the tangent at P. N.B. — In the figure the conic has been drawn an ellipse and the point lias been taken on the inside, but it is quite as easy to draw the figure when the point is on the outside of the curve, or when the conic is a hyperbola, and the point either internal or external. APPENDIX. 163 77. Def. The straight line KG' is called the polar of the point 0. If the point is on the concave side of the conic, the polar is entirely external. When the point is on the convex side of the curve, the polar intersects the conic in two points, and is identical with the chord joining the points of contact of the pair of tangents drawn from 0. Tor if the chord OQQ' drawn through the external point move round the point until Q, Q' coincide and OQQ' be- comes a tangent, the points K, Q, Q' , will evidently all coincide with the point of contact of the tangent drawn from 0. Hence the points of contact of the tangents drawn from are on the polar, and the polar is identical with the chord joining the points of contact. 78. When the pole is given the mode of constructing geo- metrically the polar, or, when the polar is given, of finding the pole is evident at once from the above proposition, whether the curve be a parabola or a central conic. Prop. VIII. If the polar of pass through o, then the polar of o passes through 0. The points and o may be either both external, or one in- ternal and the other external, but they cannot be both internal; for the polar of an internal point is wholly external, (1) Let be external. Then the polar of is the chord joining the points of contact of tangents drawn from ; and since this chord passes through o, and is the point of inter- section of the tangents at the extremities of this chord, is evidently on the polar of o. (2) Let be internal. Then since the point o is on the polar of 0, the chord of contact of the pair of tangents drawn from o, i.e. the polar of o, will pass through the point 0. COR. From this it is evident that the point of intersection of two polars to a conic is the pole of the line joining the two poles. Prop. IX. Any chord of a conic is divided harmonically by any point upon it, and the point where the polar of this point meets the chord. 164 APPENDIX. Let QQ' be any chord of a conic, and any point upon it. (1) If the point is external this proposition is already proved by Prop. VI. (2) If the point is on the concave side of the conic, let the polar of intersect QQ' or QQ' produced in 0' ; then since the point 0' is on the convex side of the curve, and is on the polar of 0, the polar of 0' i.e. the chord joining the points of contact of the pair of tangents drawn from 0' must pass through 0, and therefore as before QQ' is divided harmonically in and 0'. 79. If any number of points be on a straight line, their polars with respect to any conic will pass through the same point, viz., the pole of the straight line on which the points all lie ; and conversely if any number of straight lines pass through a point, their poles all lie on the same straight line, viz., the polar of the given point. This is briefly expressed by saying that if any number of points are collinear, their polars are con- focal, and conversely if any number of lines are cov focal, their poles are collinear. If a point instead of moving upon a straight line trace out a curve, the polars of the various points of the curve with re- spect to any conic (which in this case is called the auxiliary conic) will by their ultimate intersections form a new curve to which they are all tangents. Thus if P and Q be two contiguous points on the original curve and p and q represent their polars (with respect to any auxiliary conic) the intersection of the straight lines p and q will ultimately be a point on the new curve ; and since this point will be the pole of PQ which when P and Q are close together is ultimately a tangent to the original curve, it is evident that the polars of the several points of the new curve are tangents of the original curve, which can therefore be derived from the second curve in exactly the same manner as the second was derived from the first. On account of this reciprocal property the locus of the ultimate intersections or the envelope of (i.e. the curve touched by) the polars of the various points of a given curve, is called the polar reciprocal of the proposed curve. In analytical treatises it is shown that the polar reciprocal of any conic (with respect to an auxiliary conic) is also a conic. APPENDIX. 165 In this brief notice we shall confine ourselves to shewing that any conic section may be produced by reciprocating, i.e. taking the polar reciprocal of a circle with respect to an auxiliary circle. The advantage of having a circle for the auxiliary conic consists in the fact that the straight line drawn from any point to the centre of the circle is perpendicular to the polar of the point, and therefore also that the angle subtended at the centre of the circle by any two points is equal to one of the angles contained by the polars of the points. In finding the polar reciprocal of a given curve, there are two ways in which we may proceed. "We may either find the envelope of the polars of the different points of the original curve, or we may find the locus of the poles of the tangents at the different points of the original curve. The latter is the method which will be followed in the next article. Prop. X. The polar reciprocal of a circle, with respect to an auxiliary circle, will be a conic. Let S be the centre of the auxiliary circle, and the centre of the circle to be reciprocated. 166 APPENDIX. Join SO, and on SO produced, if necessary, take SO' a third proportional to SO and the radius of the auxiliary circle. Through 0' draw 0' M at right angles to SO', then O'M is the polar of (Art. 78). At any point P on the circle whose centre is 0, draw the tangent P Y ; and from S draw S Y perpendicular to PY. On S Y produced, if necessary, take a point P' so that SY . SP' = SO . SO' Then P' is the pole of PY. Draw P' M perpendicular to O'M; then since SY . SP' = SO . SO' .-. SP' : SO' :: SO : SY ISTow since the quadrilateral figures SP'MO', SOPY are equiangular and SP : SO' :: SO : ST it is at once seen that these figures are also similar. .;. SP' : P'M :: SO : OP .'. the locus of P' is a conic whose focus is S, directrix O'M the polar of 0, and eccentricity the ratio of SO : OP. Con. Since the eccentricity of the conic depends only upon the ratio of SO : OP, the polar reciprocal will be an ellipse, parabola, or hyperbola, according as S is inside, on, or outside the circle to be reciprocated. The distance of the directrix from S can be made as large or small as we please by increasing or diminishing the radius of the auxiliary circle, without altering the eccentricity of the curve. If the point P be such that the tangent pass through S, the point P' will be at infinity. If therefore tangents be drawn irom S to the given circle, the lines drawn through /Sat right angles to these tangents will meet the curve at infinity, and will therefore be parallel to the asymptotes of the conic, which in this case will be an hyperbola since S is outside the given circle. The focus S : the distance SO' of the directrix, and the ArPENDIX. 107 eccentricity being known, all the other elements of the conic can be at once determined. 80. By combining the harmonic properties of the complete quadrilateral with the harmonic property of the conic, viz., that any chord is divided harmonically by the pole and polar, many important theorems with regard to figures inscribed in and circumscribed about a conic may be deduced. Prop. XI. The triangle formed by the point of intersection of the diagonals of a quadrilateral figure inscribed is a conic, and the points of intersection of the opposite sides is self-conjugate, i.e. each angular point is the pole (with regard to the conic) of the opposite side. Let ABGD be a quadrilateral inscribed in a conic. Now since the chord BO is divided harmonically in F and S, {Prop. V.) .'. the polar of F passes through S. So the polar of F passes through F. . \ FS or FG is the polar of F. 168 APPENDIX. So also EF is the polar of G. Hence E the point of intersection of EF and EG is the polar of FG and .*. the triangle EFG is self-conjugate. Prop. XII. The intersection of the diagonals of a quadrilateral figure described about a conic is the pole with respect to the conic of the straight line joining the points of intersection of the pairs of opposite sides. A & B G Let ABCB be a quadrilateral figure described about a APPENDIX. 169 conic ; then the intersection E of the diagonals A G y BD shall be the pole of the straight line FG joining the points of intersection of AD, BG and AB, DG. Let a, b, c, d be the points of contact of the sides Of the quadrilateral with the conic. Complete the quadrilateral abed, and let e be the inter- section of the diagonals ac, bd; and / and g those of the pairs of the opposite sides ad, be and dc, ab respectively. Now by the last proposition e is the pole of fg with respect to the conic which is inscribed in A BCD and therefore described about abed. But since F is the pole of db and G is th 3 pole of ac .'. e is the pole of FG . ' . the straight lines fg and FG coincide. Again since A is the pole of a d and G is the pole of be .'. /is the pole of AG So also g is the pole of BD .-. E is the pole oifg .'. the points E and e coincide, but e is the pole of FG .'. E is the pole of FG. Cor. 1 . 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Extra fcap. 8vo. $s. EUROPEAN HISTORY. Narrated in a Series of Historical Selections from the Best Authorities. Edited and arranged by E. M. Sewell and C. M. Yonge. First Series, 1003— 1 154. New Edition. Crown 8yg. 6s. Second Series, 1088 — 1228. New Edition. Crown 8vo. 6s. LONDON: R. CLAY, SONS, AND TAYLOR, WINTERS. V /v -riTTT'" "•"»/"»< ; ; 4 800571 UNIVERSITY OF CALIFORNIA LIBRARY v ■; WW, &■ HHHHHH ■"' '■'''•• •. '■:'':; IBM UP' Ih2 IMPl m : iSSBli ;■-'■■■.■ .ay ■■''• i&ffl» H 1 sun -•''■'- .■•■ .