u r,^ GEOMETRICAL CONIC SECTIONS, AN ELEMENTARY TREATISE IN WHICH THE CONIC SECTIONS ARE DEFINED AS THE PLANE SECTIONS OF A CONE, AND TREATED BY THE METHOD OF PROJECTIONS. BY J. STUART JACKSON, M.A., LATE FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE. MACMILLAN AND CO. 1872. [All Rights reserved.} 3"3 Camfttitise : PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. PREFACE. The following pages have been written with a view to give the student the benefit of the Method of Projections as applied to the Ellipse and Hyperbola. This Method is calculated to produce a material simplification in the treatment of those curves and to make the proof of their properties more easily understood in the first instance and more easily remembered. It is also a powerful instru- ment in the solution of a large class of problems relating to these curves. When the Method of Projections is admitted into the treatment of the Conic Sections there are many reasons why they should be defined, not as has been the case of late years with reference to the focus and directrix, but according to the original definition from which they have their name, as plane Sections of a Cone. First and principally, because this definition gives an immediate proof of the relation by projection of the Ellipse and Circle and of the General and Rectangular Hyperbolas : and in the second place, it naturally divides the properties that may be proved by projection from those connected with the focus and directrix, and thus introduces a valuable simplification into the treat- ment of the subject. It is also a consideration of some importance that we can see at once from the form of the cone, the general form of the curves that may be cut from 357855 IV PKEFACE. it by a plane in different positions ; and, by turning the plane about a certain line, we see how the curves pass from one form into another. It is hoped these may be thought sufficient reasons for departing from the definition which has been in use of late years. I have retained the proof of the constancy of the ratio of the rectangles under the segments of intersecting chords in constant directions (Chap. v. § 15), given by Dean Hamilton in his treatise published in 1773, as an improving and interesting study and one that is in harmony with the scheme of this work. I have to acknowledge the great kindness of Rev. Chas. Taylor, Fellow of St John's College, Cambridge, in per- mitting me the use of his excellent proof of the intersection of tangents at the extremities of a chord of any Conic Section on the diameter of the chord; with the proofs of GV. GT = GP^ (Chap. V. § 8), and PV=PT (Chap. vii. § 20), depending on it : and also of his proof of 8QP= HQP' (Chap. IV. § 13). Professor Adams' property of the tangent is indispensable in any geometrical treatment of the Conic Sections, and I have his kind permission to make use of it. I shall be greatly obliged by any corrections and hints towards improvement in case this work should be so fortu- nate as to reach a second edition. 9, Bbookside, Cambridge, Christmas, 1871. TABLE OF CONTENTS. CHAPTER I. ON THE METHOD OF PROJECTIONS. ART. PAGE 1. Definitions 1 2. Construction 1 3 — 5. Relation between lines and their projection ... 2 6. Effect of projection on angles 4 7. „ „ „ areas 5 8. Tangents project into tangents 5 9. Object of the Method 6 CHAPTER II. ON THE CONE AND SPHERE. 1, 2. Definitions, &c 7 3. Every plane section of a sphere is a circle .... 8 4, 5. Tangents to a sphere from an external point are equal . 8 CHAPTER III. ON THE ELLIPSE. 1. Definition 10 2. The ellipse is the projection of a circle 11 3. Minor axis 12 4. Conjugate diameters 13 5. Supplemental chords 14 6. QF^:CD'::PV. Vp.CP^ 14 VI TABLE OF CONTENTS. ART. PAGE 7—9. CV.CT = CP^'.CN.CT^ GA\hc. . . . . 14 10. PF.PG = BG^',PF.Pg = AC"" 15 11. NG : NO :: BC^ : AC^ 16 12. GP^ + <7i)2 z. AG^ ■\- BG^ ' . . 16 13. QO . Gq : RG . Gr a constant ratio 17 14. Circumscribing parallelogram 18 CHAPTER IV. FOCAL PROPERTIES OF THE ELLIPSE. 1,2. Preliminary 24 3 — 5. Foci and directrices . 25 6. Secant and tangent 29 7. Tangent equally inclined to the focal distances ... 30 8. Auxiliary circle 31 9. SY.HZ = BG^ 32 10. SP.HP=GD^ . .33 11. ST : QN :: AS : AX 33 12 — 14. Intersecting tangents 34 m SG '. SP V. SA . AX 36 CHAPTER V. ON THE HYPERBOLA. 1. Definition 42 2. Projection of the rectangular hyperbola 43 3. Magnitude of the axes 46 Properties of the rectangular hyperbola : — 4. Asymptotes 47 5. Intercepts between asymptotes and curve equal. Locus of the middle points of parallel chords 48 6. QV^ + GP"" = VU^ 61 7,8. GV.GT = GP^i GN.GT = GAK . . . . 52 9. Conjugate hyperbola, conjugate diameters .... 64 10. Inscribed parallelogram 56 11. Intercepts on a line parallel to an asymptote ... 57 TABLE OF CONTENTS. Vll ART. PAGE 12, 13. These properties transferred to the general hyperbola . 68 14. Supplemental Chords 61 15, 16. Ratio of rectangles on the segments of intersecting chord 61 17. NG : NC :'. BC^ : AC^ 65 18. PF.PG = BC^, PF.Pg = AC^ 65 19. Asymptotes parallel to two generating lines of the cone . 66 CHAPTER VI. FOCAL PROPERTIES OF THE HYPERBOLA. (See contents of Chapter IV.) CHAPTER VII. ON THE PARABOLA. 1. Definition. 87 2. Limiting form of semi-ellipse or semi-hyperbola ... 88 3. NP^ = 4:AS.AN 89 4. QR.Q'R = 4AS.PR 90 6. Locus of middle points of parallel chords . . . .91 6. QR^ = 4AS .PV 92 7 — 9. Focus and directrix 93 10. Secant and tangent 96 11. Tangent equally inclined to the focal distance and the axis of the curve 96 12. Tangents at the extremities of a focal chord .... 97 13. Perpendicular from the focus on the tangent . ... 97 14—18. Intersecting tangents 98 19. Tangents at the extremities of a chord intersect on its diameter . . . 99 20. PT=PV 99 21. QV^^4.SP.PV 100 22. Length of focal chord . . . , . . . .101 23. Ratio of the rectangles under segments of intersecting chords 101 24,25. The normal 102 EEEATA. Two lines from the bottom of p. 61, after rectangles insert under the segments. p. 72, for Chap. VII. read Chap. VI. p. 87, for Chap. VIII. read Chap. VII. CONIC SECTIONS. CHAPTER I. On the Method of Projections. 1. Definitions. The Orthogonal projection of a point on a plane is the foot of the perpendicular drawn from the point to the plane. The plane on which the projection is made is called the plane of projection. The orthogonal projection of a line on a plane is the line traced on the plane by a straight line which passes through all successive points of the given line and is always perpen- dicular to the plane of projection. In the present treatise when projections are spoken of it may be understood that orthogonal projections are intended. If a straight line be drawn through two adjacent points of a curve, and the two points move towards one another on the curve until they coincide, the straight line in its final position is said to be a tangent to the curs^e at the point which it has finally in common with it. 2. In order to project a straight line AB (fig. § 5) on a plane ahc, we must draw the perpendicular Aa from some J. c. s. 1 2 CORIC SECTIONS. point A of AB to tke plane of projection (Euclid xi. 11), then make a plane pass through AB, Aa ; it will be perpen- dicular to the plane of projection (xi. 18). Hence the per- pendicular drawn from any point P of AB to ah the inter- section of the planes will be perpendicular to the plane of projection ; and the straight line ah will be the projection of AB, Hence the projection of a straight line is also a straight line. 3. The projections of parallel straight lines are also parallel. From points A and C (fig. § 5) in the parallel straight lines AB, CD draw Aa, Cc perpendicular to the plane of projection, these lines will be parallel (Euclid xi. 6), and the planes BAa, DGc will be parallel (xi. 15) : also the plane of projection will cut these planes in parallel lines ah, cd (xi. 16), and these are the projections oi AB, CD. Hence the pro- jections of parallel straight lines are parallel. 4. A line of finite length and its projection are cut in the same ratio by any point and its projection. Let a, p, h be the projections of the points A, P, jB in the straight line AB (fig. § 5), draw AE perpendicular to Bh cutting Pp in Q, then Ap, Qh are parallelograms, and AQ, QE equal to ap, ph. Also PQ is parallel to BE the side of the triangle BAE, and therefore AQ : QE :: AP : PB, or ap : ph :: AP ; PB. The line AB and its projection a5 are cut in the same ratio by the point P and its projec- tion p. 5. Parallel straight lines of finite lengths are diminished by projection in the same ratio. Let the parallel straight lines AB, CD of finite length have projections ah, cd. From A and G draw AE, OF perpen- dicular to Bh, Dd : these lines are equal to ah, cd : they are ON THE METHOD OF PROJECTIONS. S also parallel to these parallel straight lines and are there- fore parallel to one another (Euclid xi. 9). The lines AB, CD are also parallel, hence the angles BAE, BCF ave equal (xi. 10) : the angles at E and F are right angles, and the triangles BAE, JDCF are similar. Hence BA : AE :: DC : CFov BA : ah :: DC:cd) AB, CD are diminished by projection in the same ratio. Hence it appears that the ratio in which any line is altered by projection from one plane to another depends only upon its inclination to the line of intersection of the planes. It may be seen that lines parallel to the line of intersection are unaltered by projection, and that lines more inclined to it are diminished in a greater ratio than those less so : while lines at right angles to the line of intersection are diminished in the greatest ratio. The ratio which a line at right angles to the line of intersection of its plane with the plane of pro- jection bears to its projection may conveniently be called the maximum ratio for the two planes, and it will increase from unity to a ratio as great as we please when we increase the inclination of the planes from zero to a right angle. 1—2 4 CONIC SECTIONS. In other words, a line at right angles to the line of inter- section may have its projection as nearly equal to itself or as small as we please by taking the plane of projection at first very slightly inclined to the plane of the line, and then in- creasing the inclination till it becomes a right angle. The projection is the base of a right-angled triangle of which the line is the hypotenuse, and the inclination of the planes the angle at the base. By increasing the angle at the base from zero to a right angle we diminish the base from the length of the hypotenuse to zero. 6. Angles will be in general increased or diminished in magnitude by projection, except those contained by lines parallel and right angles perpendicular to the line of in- tersection. Of the four angles contained by two intersecting lines, that which is subtended by the line of intersection and the vertical opposite angle will be increased, the other two di- minished by projection ; and these angles may be increased and diminished to any extent by making the inclination of the plane of projection suffi- ciently great. If, for example, it be required to increase the acute angle ACB subtended by AB the line of intersection to a right angle ; on AB de- scribe the semicircle AcB, and draw CD perpendicular to AB cutting the semicircle in c. Then let the inclination be so chosen that the maximum ratio shall be CD : cD : then CB will have its projection equal to cB, and the angle A CB will project into an atigle equal to AcB, i.e. to a right angle. ON THE METHOD OF PKOJECTIONS. 5 This only holds, however, when D lies between A and B. If D fell outside AB, the angle ACB might be increased or diminished by projection, and would be greatest when the circle circumscribing ABG touched GDy and in that case DC would be a mean proportional between DA and DB. 7. The area of any figure will be reduced by projection in the maximum ratio of the planes. For every triangle may be divided by a line through one of the angular points parallel to the line of intersection into two, each having its base parallel and its altitude at right angles to the line of intersection : the base of each will be unaltered by projection, the altitude diminished in the maximum ratio. Hence the area of each and therefore of the two together will be diminished in the maximum ratio. Hence, whatever its magnitude or position, every triangle is diminished by projection in the maximum ratio of the planes. But every figure, whether rectilinear or curvilinear, may be as nearly occupied by triangles as we please, by making them sufficiently numerous. Each of these will be diminish- ed by projection in the maximum ratio of the planes; and therefore the figure composed of them will be diminished in the same ratio. 8. The projection of the tangent to a curve at any point is the tangent to the projection of the curve at the pro- jection of the point. If pf q be the projections of two points P and ^ of a curve, the straight line pq is the projection of the straight line PQ ; and if p, q move so as to be always the projections of P and Q as they approach one another on the curve until they coincide and Q is merged in P, p and q will also approach one another on the projected curve, and will finally 6 CONIC SECTIONS. coincide in the projection of P, PQ then becomes the tangent to the curve at P, and its projection pq becomes the tangent to the projection of the curve at the projection of P. 9. The object of the method of projections is to extend our knowledge of curves from those which we already know, or can easily investigate, to others into which they project. Thus the properties of the ellipse can be deduced from those of the circle, and the rectangular hyperbola helps us to ascertain the properties of the hyperbola of unequal axes. The preceding propositions serve to connect the diameters and other lines, and the areas, of the known curve with those of its projection. Examples. 1. If the inclination of two planes be the third of two right angles, and a line of 4 yards in one of the planes be inclined to the line of intersection at half the above angle, find the area of the square on the projected line. 2. In Art. 6 when D falls without AB^ find the inclination of the planes in order that, in any given case, the angle ACB may not be altered by projection. 3. A triangle is projected on to a plane whose line of inter- section with its own plane coincides with the base ; it is projected back again on to its original plane, and the inclination of the plane is such that after these two projections its vertical angle is unalter- ed. Shew that this inclination gives the greatest possible increase to the vertical angle at the first projection. 4. Find the altitude of the sun when the greatest shadow of the side of a square on a horizontal plane equals its diagonal. 5. In what position must a cube be placed so that its shadow on a plane that receives the sun's rays directly may be the greatest possible? CHAPTER 11. On the Cone and Sphere. 1. Definitions. If through the centre of a circle ABO we draw a straight line F at right angles to its plane, then the surface traced out by a straight line which passes through any point V of this line and the successive points of the circle ABG is called a Cone; or, more distinctively, a Bight Circular Cone, OF is called the Axis of the cone, and Fits vertex. We repeat here Euclid's definition of a sphere : A sphere is a solid figure described by the revolution of a semicircle about its diameter which remains fixed. 2. If VP be any position of the generating line of a cone, the angle OFP is the vertical angle of a right-angled triangle of which the height OF and the base OF are con- stant ; therefore OFF is constant, and the angle between the axis and the generating line in all its positions is invariable. Also if any plane at right angles to the axis cuts the cone in the curve ahc, ahc will be a circle. Let o and p be the points 8 CONIC SECTIONS. in which the plane cuts the axis and the generating line VF ; join op. Then op is the base of a right-angled triangle of which the height o V and the vertical angle o Vp are invariable for all positions of VF; hence op is constant, and the section ahc is a circle, with centre o. Every plane containing the axis cuts the cone in two right lines inclined to each other at a constant angle. 8. From the definition of a sphere it is evident that every point in the sphere is equidistant from the centre of the generating circle. Every section of a sphere by a plane is a circle. Let F be any point of the section AB (7 of a sphere whose centre is made by a plane. Draw ON at right angles to the cutting plane, and join OF, NF. Then for every point in the section the height ON and the hypotenuse of the right-angled triangle ONF are constant, and therefore the base NF is constant, and ABC is a circle of which N is the centre. 4. Definition. A straight line is said to touch a sphere when it touches the circle in which the plane through the line and the centre of the sphere cuts the sphere. It is evident that if a plane be drawn at right angles to the diameter of a sphere through its extremity, every line drawn in this plane through the extremity of the diameter ON THE CONE AND SPHERE. 9 will touch the sphere, and therefore the plane is said to touch the sphere. 5. AH the lines drawn from one point to touch a sphere are equal. Let TP (fig. § 3) be any line drawn from the point T to touch the sphere ABGD whose centre is in the point P. Join OP. Then TP touches a circle of which OP is a radius, therefore OPT is a right angle. Join OT, Then TPisthe base of a right-angled triangle of which the height OP and the hypotenuse OTsne constant. Therefore all the lines drawn from T to touch the sphere are equal. Also the angle OTP is constant, and the equal tangents will therefore generate a cone of which T is the vertex and OT the axis. Also the points common to the sphere and cone lie on a circle whose plane is at right angles to OT, CHAPTER III. Gn the Ellipse. 1. Definitiok JjQi AVA' he a cone whose axis VO is in the plane of the paper, and VA, VA' its generating lines in that plane. And let the cone be cut by a plane APA' at right angles to the plane of the paper and intersecting it in the line A A' such that A, A' are points in the generating line on oppo- site sides of the axis VO ; then the section APA is call- ed an Ellipse. It is manifest that the ellipse will be a closed curve, and divided into equal parts by the line AA! : every line in the plane of the section drawn at right angles to AA' wiU meet the curve in two points on opposite sides of AA j and at equal distances from it. A A is called the axis major of the ellipse, and a line through C its middle point in the plane of the curve at right angles to AA limited by the surface of the cone is called the axis minor. ON THE ELLIPSE. 11 2. Every ellipse may be considered as the projection of a circle whose diameter equals the axis major of the ellipse. Let DBD', BPR be two circular sections of the cone made by planes at right angles to the axis through (7, N, the middle and any other point of AA\ and cutting the plane of the curve in BGB\ PNP' at right angles to AA'. Then JVP^ = RN. NR and BC' = DO. GD\ Also RN : DG :: AN : ^(7 and NR' : GD' :: NA' : A'G\ .-. RN. NR : DG. GU :: AN, NA' -, AG\ :. NP' : BG' :: AN.NA' : AG'. Now if NQ be the ordinate of a circle drawn on A A' as diameter, we shall have NQ^ = AN. NA\ and NP^ :BG':: NQ" : A C^ or NP: BG ::NQ :AG, NP'.NQ ::BG :AG; 12 CONIC SECTIONS. then if a circle, diameter AA', be inclined to the plane of the paper at such an angle that the maximum ratio of diminution of the planes be AG : BC, every ordinate of the circle will be diminished by projection in this ratio, and will there- fore after projection equal the corresponding ordinate of the ellipse, and the circle whose diameter is A A' will project into an ellipse whose axes equal AA', BB\ 3. We must now prove that in every case the diameter BB' of the ellipse at right angles to the plane of the paper is < A A' in this plane. BC' = I)G.CI)': vfe have to shew that DC . CD' < AG\ Draw AE perpendicular to DD\ and AF parallel to D'A' so that CF will = CD\ Then CA^^CE'^ + AE^ = DC. CF+EF' + AE' (Euclid ii. 6) = DC.CD' + AF' {or AD'), thus DC, CD' is always < CA\ and the ratio BC : A Gone of less inequality. We may now trace some of the properties of the circle in the ellipse by means of the principles of projection already proved. ON THE ELLIPSE. 13 4. In the circle if two diameters are at right angles to one another each bisects all chords parallel to the other. But by Chap. I. § 4, the projection of a point which bisects a line bisects the projection of the line; therefore chords of the circle which are bisected by a diameter will project into chords of an ellipse bisected by the projection of the dia- meter. Hence diameters of the circle at right angles to one another project into diameters of the ellipse which bisect each the chords parallel to the other. These diameters of the ellipse are said to be conjugate on account of the relation between them being mutual. Also the tangents to a circle at the extremities of a diame- ter are at right angles to the diameter and parallel to the chords bisected by it : and the parallelism of straight lines is not destroyed by projection ; hence the tangents at the two extremities of a diameter of an ellipse are parallel to the chords bisected by it and to the conjugate diameter. Those angles at the centre contained by conjugate diame- 14 CONIC SECTIONS. ters in which the axis major lies will in all cases be less than a right angle. The accompanying figiire shews how diameters of the circle at right angles to one another project into conjugate diameters of the ellipse. 5. Supplemental Chords. The angle in a semicircle is a right angle in whatever point of the circumference the chords containing it meet : therefore when projected these chords are parallel to conjugate diameters of the ellipse : such chords when projected are called supplemental chords ; they form two sides of a triangle the base of which is a diameter, and its vertex on the circumference of the ellipse. Hence supplemental chords are parallel to conjugate diameters. 6. If Q F be a semi-chord of a circle at right angles to a diameter PCp, and CD the radius parallel to QV, then QV^=^FV. Vp ; or since CP, CD are equal, QV : CD-" :: PV. Vp : OP". These ratios are compounded of the ratios of parallel lines, and are therefore unaltered by projection, therefore also in the ellipse QV : CD' :: PV. Vp : CP\ Definition. The half of a chord bisected by a diameter (as QVhj PCp) is called an ordinate to that diameter. 7. If Q Ty the tangent to the circle at Q, meet CP pro- duced in T, CV.CT^ CQ^ or= (7P^ or CP is a mean proportional between V and T. The ratios of these lines are not altered by projection ; therefore also in the ellipse GV, CT=: CP\ As a particular case of this proposition, if the tangent at P meets the axis major in T, and P^be the ordinate, CN. CT:= CA\ ON THE ELLIPSE. 15 8. The tangents to a circle at the extremities of any chord meet in the diameter (produced) which bisects the chord: and the tangent at the extremity of the diameter when terminated by the pair of tangents is bisected at its point of contact. These properties are not affected by pro- jection, and therefore hold true in the ellipse. Cor. Hence to draw tangents to an ellipse from any point T, join OT and let it cut the curve in P; draw the tangent at P by the latter part of the last Art. In CP take a point V such that CF is a third proportional to GTy CP. Through V draw an ordinate QVq parallel to the tangent at P: join TQ, Tq, then by the proposition of the last Art. these will be the tangents required. 9. The above propositions might equally have been made to take the form, if Qv be the ordinate to (7i>, and the tangent at Q meet CD in t, Qv' : CP' :: Pv . vd : CD\ and Cv . Ct = GD\ As a particular case of the latter, if Pn be the ordinate from P to the axis minor, and the tangent at P meet the axis minor in ^, Cn.Ct=GB\ 10. PF.PG = BC\ PF.Pg^AC\ Let the normal at P {the line through Pat right angles to the tangent) meet the axes and jD Gd the diameter con- jugate to CP in G, g and F-, and let the ordinates PN, Pn produced meet Dd in R, r. Then in the quadrilateral GFEN, the angles at F and N are right angles ; and the quadrilateral may be inscribed in a circle. 16 CONIC SECTIONS. Hence FG.PF=:PN.PE=: Cn.Ct = BC\ Similarly a circle on rg as diameter will pass through n 2.n& F, ^nd. PF . Pg = Pn . Pr = GN .GT= AG\ 11. NG : NG :: BG' : AG\ If we draw a circle on the axis major of the ellipse as diameter it is called the auxiliary circle, and will be equal to that which projects into the ellipse ; and if the projected circle were made to revolve about its diameter AA', it would come to coincide with the auxiliary circle: and the point which, when the circle was projected, projected into P will now be found in the ordinate NP produced, and if this point be called Q we shall have NP : NQ :: BG : AG. Also the tangent to the circle which projects into the tangent PT passes through T, and when made to revolve about AA' will come into the position QT: hence the tangent to the auxiliary circle at Q will meet AA produced in the same point as the tangent to the ellipse at P. Hence we shall have NG , NT:= NP' and NG . NT= NQ' ; .-. NG : NG :: NP' : NQ' :: BG' : AG\ 12. By the reasoning of the last article we see that if CP, GD be con- jugate semi-diameters, Q, R the points in which the ordinates NP, MD produced meet the auxiliary circle, QGR will be a right angle. ON THE ELLIPSE. 17 Hence the triangles GQN, RCM will be equal in all respects. Hence GM= QN, and ON' + CM' = ON' + QN'= CQ' = A C\ Also QN= CM, BM= CN; /. QF'+RM'^^AC^ and FN'+mP : QN'+RM' ::BC':A C; .-. FN' + DM' = BC\ hence CP' + CD'= CJT + CJ\P + FN' + DM^=: AC' + BC\ 13. In the circle if the chords Qq, Br intersect in 0, we have QO , Oq — BO, Or; and if we draw radii (7P, CD parallel to the chords we have QO.Oq: CF'y.BO.Or: CD\ These ratios are not affected by projection, and the same proportion holds in the ellipse: it is independent of the position of 0, but depends only on the direction in which the chords are drawn. It is equally true if be a point exterior to the ellipse. And if tangents be drawn to an ellipse from any point, their lengths will be proportional ta the parallel semi-diameters. Hence if a circle intersect an ellipse in four points and the common chords be drawn, it will easily appear that the J. C. 8. 2 18 CONIC SECTIONS. semi-diameters to which these chords are parallel must be equal, and therefore they and the chords parallel to them equally inclined to the axes. 14. The area of the parallelogram circumscribing an ellipse and touching it at the extremities of conjugate diameters ==4, meeting it in /, and the parallel tangent in J, Shew that / and J trace similar curves, 28. If P be any point of an ellipse, and AP, A'P produced meet the tangents at A'y A in R and S, the tangent at P will bisect AS and A'R. 29. If from an external point 0, two straight lines OAP^ OQA' be drawn through the vertices of an ellipse APA!Q\ if QA, A'P intersect in i?, OR is at right angles to the axis majqr. 30. If TP, TP' be tangents to an ellipse, and PC'p be the diameter through P, then P'jp is parallel to CT. 31. If TP^ TP* be tangents to an ellipse from an external point T, TR the diagonal of the parallelogram on TP^ TF and R be on the ellipse, then T will lie on an ellipse similar and similarly situated to the former. 32. If from any point T exterior to an ellipse, a line be drawn parallel to either axis to meet the curve the first or second 22 EXAMPLES. time in Q, tlie line bisecting TQ at right angles and that bisecting the tangents from T will meet on the tangent at Q. 33. Find the centre of a given ellipse. 34. Find the axes of a given ellipse. 35. In a given ellipse find the diameter conjugate to a given diameter. 36. If two points of a rod be constrained to move in two fixed lines which intersect at right angles, every other point of the rod will describe an ellipse. 37. If from any point F of an ellipse FQ be drawn to the axis major equal to jB(7, then if FQ produced either way meets the axis minor in F, FR = AG. 38. If FCF', BOB' are conjugate diameters, and FQ is drawn parallel to the axis major to meet the curve in Q\ prove that DQ\^ parallel to two of the lines joining extremities of the axes of the curve. 39. If NF produced meets the auxiliary circle in Q^ prove that GF, GQ produced meet on a circle whose diameter = sum of the axes of the curve. 40. Two ellipses have their axes equal each to each and in the same plane, also their centres coincident, draw the common tangents. 41. If (7-4, GB be any conjugate diameters of an ellipse and CB be produced to any point B\ and an ellipse is described on GAy GB' as conjugate diameters; if the ordinate F'FN be drawn parallel to BG, shew that the tangents to the ellipses at P, F' will intersect at a point lying on GA produced, and that FF : FN :: BG : BG, EXAMPLES. 23 42. If two ellipses with major axes parallel or at right angles intersect in four points, the opposite sides of the quadrilateral formed by joining the four points will be equally inclined to either axis of either curve. 43. If any system of diameters of an ellipse of a given even number divide it into equal sectors, the sum of the squares on the diameters is the same whatever their directions. 44. The same when the number is odd or even. 45. Prove that P6«.P^ = Ci>'. 46. If GR be the perpendicular from the centre on the tan- gent at P, and BR, AD be joined ; prove that the triangles ACD, RGB are similar. CHAPTER lY. Focal Properties of the Ellipse. 1. We have now to shew that there are two points within the ellipse which, regarding the curve in its relation to the circle, are as it were a divided centre, and also two lines exterior to it which bear a remarkable relation to these points and the curve. 2. Let us premise the following propositions : If the circle inscribed in the triangle VAA' touches the sides at K, S, L; and the circle that touches AA' and VA, VA' produced touches them at II, K\ L\ then KK' = AA, aLndAS=-A'K For the perimeter of triangle VAA' r=:VA + AH+ffA' + A'V = F^ + AK' + LA' 4- VA' = VK' + VL' = 2VIC = 2VK+2KK': also =^VK+KA + AA' + A'L + LV = VK+AS+AA' + A'S + LV=2VK+2AA\ ,\KK' = AA\ Also KK' = KA + AK' = AS+ AH, /. AA' = AH+AS: hut A A' = AH + A' E, .-. A8=A'H. FOCAL PEOPERTIES OF THE ELLIPSE. 25 Also if we produce AA' both ways to meet LK, K'Ll produced in X, X\ AX will = AX'. For AK, A 'L' being equal to A 8, A'H are equal, and being equally inclined to the axis of the cone will have equal projections on it. And the projections of AXy AX' will equal those of AK^ A'L\ and are therefore equal: therefore AX, A'X', being parts of the same line with equal projee-, tions on the axis of the cone, are equal. 3. Now to return to the construction in Chap. III. § 1. Let AA' be the intersection of the plane of the paper with a plane at right angles to it that cuts the cone VAA' in the ellipse APA'. Inscribe in the triangle VAA' the circle SKL with centre on the axis of the cone, and escribe the circle HK'L' with centre 0' also on the axis. Then if we make the circles to revolve about their diameters which coincide with the axis of the cone, they will generate spheres which will touch the cone in the circles KRL, K'R'L'. Every point of each of these circles is equidistant from F, and therefore the distance RE from one circle to another along a generating line of the cone will be invariable. Also OS, O'lTwill be at right angles to the cutting plane, which will therefore touch the spheres in S and H, 26 CONIC SECTIONS. Let P be any point of the elliptic section, draw VBPB' the generating line of the cone through P, and join 8P, HP. Then PS, PR are drawn from P to touch the sphere centre in 8 and i?, therefore PS==PB, and PH, PE touch the sphere centre O m H and R\ therefore PII=^ PR\ Hence SP + EP=PB + PPf = PE=:KK'=^AA\ Hence the sum of the distances of any point of the ellipse from the points Sj H within it is invariable and ^-AA\ Hence we obtain the following construction to enable FOCAL PROPEETIES OF THE ELLIPSE. 27 US to describe an ellipse: Fasten the two ends of a thread to the two points S and H, and let the thread be longer than SH: then stretch it with the point of a pencil, and mark the line which is traced by moving the pencil on all sides of the points so as to keep the thread tightly stretched: the curve so traced will be an ellipse. Viewing the ellipse in its relation to the circle, ;S^ and H may be considered as a divided centre, the sum of the distances of all points in the circumference from them being the same. 4. Now let the plane through P, perpendicular to the axis of the cone, cut the cone in the circle QPQ', and the plane of the ellipse in NF at right angles to AA\ Then we shall have SP=PB=QK, and the triangles QAN, KAX are similar; .-. QK : NX :: AK : AX :: AS : AX, Hence whenever P is situated on the ellipse, SF : NX is a constant ratio = ^>Sf : AX. And similarly, joining HF, HF{=FE' = Q'L') : NX r. A'H : A'X\ The ratios AS : AX, A'H : AX' are equal, and since A'L' < A!X', each is a ratio of lesser inequality : either of them is called the eccentricity. 28 CONIC SECTIONS. So then if we draw the elliptic section in the plane of the paper, and draw through X, X' lines XZ, X'Z' at right angles to XX\ and PM, PM' perpendiculars to these lines ; we have for any point P of the ellipse SP : NX (or PM) :: AS : AX, HP : NX (or PM') :: A'E : A'X. S and H are called the foci of the ellipse, XZ, X'Z' the directrices, 5. The position of the foci and directrices is determined by the following relations : (7/Sf',4- CB' = CA' ; FOCAL PKOPERTIES OF THE ELLIPSE. 29 For joining the foci with the extremity of the axis minor, SB==HB = i{SB + I{B) = AG,andSG' + BC' = SB' = AC\ Also CXis a third proportional to CS smd CA. For first, SB : CX, i.e. CA : CX:: SA : AX. Hence CA : CX :: SA : AX, CA : SA :: CX : AX, CA-SA : SA :: CX - AX : AX, or CS : SA :: CA : AX, .-. CS : CA :: >Sf^ : AX, :: a^ : (7X, or CXia a third proportional to CS and (7^. G. Properties of the secant and tangent. Let FSp, F'Sp' be two focal chords. Let the secant FF cut the directrix in F, join SF, Let QSq bisect the vertical opposite angles FSF, pSp: FM, FM' perpen- dicular to the directrix. Then SF : SF' :: FM : FM' :: FF : FF, therefore SF bisects the angle FSp (Euclid VI. A) and is so CONIC SECTIONS. perpendicular to Q8q\ hence also the secant pp will pass through F, Now let the secant FPP' revolve about F, so that P, F will approach one another; SQ still bisecting the angle P8P' will be perpendicular to SF and therefore constant in position ; and P, P' will finally coincide with Q which will then be a point on the curve, and the secant will then become a tangent to the curve at that point. Similarly if the secant Fpp turn about F, it will in its limiting position touch the curve in Sq produced. Hence the tangents at the extremity of a focal chord intersect on the directrix : and the part of any tangent between the curve and the directrix subtends a right angle at the focus. 7. The focal distances make equal angles with the tangent at any point. Let the tangent at P meet the directrices in F, F\ join 8F,HF'\ FSF, PHF' will be right angles. FOCAL PKOPERTIES OF THE ELLIPSE. 81 Also the triacgles MPF, M'PF' are similar, and SB : PM :: AS : AX :: HP : PM' ; .-. SP : PH :: PM : PM' :: PF : PF\ or SP : PF :: HP : PF\ Hence (Euclid VL 7) SPF, HPF' are similar triangles, and the angles SPF, HPF' are equal. Cor. The tangents at the extremities of the axes are at right angles to them. 8. The feet of the pei-pendiculars from the foci on any tangent lie on the auxiliary circle. Let SY the perpendicular from S on the tangent at P meet HP produced in K. Join SP, CY, Then in the right-angled triangles SYP, KYP, PY is common and the angle SPY= the angle HPZ (by the last proposition) = the vertical opposite angle KPY. Hence SY=KY, and SP=KP: .'. KH^ KP+PH=SP + PH= 2A C. S2 CONIC SECTIONS. And C, Y being the middle points of SH, 8K, CF is parallel to HK and half of it and therefore = ^ (7. Hence Y lies on the circle on AA' as diameter (the auxiliary circle). And similarly Z the foot of the perpen- dicular from H, Cor. 1. OF being parallel to 5Tand bisecting SH, also bisects 8P. Hence SYP being a right angle, the circle on 8P as diameter passes through F, and has its centre on CY\ hence it touches the auxiliary circle at F. Cor. 2. If CD be drawn parallel to the tangent at F and therefore conjugate to GP, and intersect PH in E, then CEPYis a parallelogram, and PE= GY^AC. 9. If SY, HZ are perpendiculars on the tangent at P from the foci S, H; 8Y. HZ=:BG\ Since YZH is a right angle, YG produced will meet ZH produced on the auxiliary circle (at Y'). The triangles 8GYj HGY' are equal in all respects, and 8Y. HZ^ EY' . HZ^ AH , A'H^ A' C' - GH' = BG'. FOCAL PROPERTIES OF THE ELLIPSE. 10. 8P,HP=^CD\ 33 If we draw PF in fig. § 8 perpendicular to DC (produced if necessary) we shall have the triangle PEF similar to HPZ and SPY\ andP£' = ^C: hence SP : SYv. HP :EZ::PJE:PF::AC: PF :: CD : DC, v PF, CD = AC.DC ; .-. SP.HPiSY. HZ:: CD"" : DC\ and 8Y . HZ^BC, .'. SP.HP=CD\ 11. If from any point Q of the tangent PK perpen- diculars QN, QT he drawn to the directrix and SP, then ST : QN :: AS : AX. For QT is parallel to KS; and if we draw PM perpendicular to the directrix, ST: SP:: QK : PK :: QN : PM, .-. ST : QN :: SP : PM :: AS : AX. The student of analytical geo- metry will see in this proposition the basis of the polar equation to the tangent in terms of the angle PSA (a), viz. p = . ""'-^.r,^^ ^ . ^ ^' ^ cos (a-^j + ecos^ The proposition is due to Professor Adams, and the property is equally true of all the Conic Sections. J. c. s. 3 34 CONIC SECTIONS. 12. Hence if QP, QP' be the tangents to an ellipse from an exterior point Q, QP, QF subtend equal angles at 8, For if we draw QT, QT, QM the perpendiculars on SP, SP' and the directrix, we shall have 8Tj 8T' in the same pro- portion to QM and therefore equal. Hence the right-angled tri- angles Q8T, Q8T' are equal in all respects, and the angles Q8P, QBF equal. If Q lie beyond the directrix, T, T will lie in P8, F 8 produced, and the angles Q8P, Q8F will be proved equal by proving their supplements equal. 13. If QPy QF be the tangents to the ellipse from Qj the angles 8QP, HQP' are equal. Produce 8P, HP' to B, B'\ and let HP, 8P' intersect in a Then QP, Q^ bisect the angles HPR, P'8P respectively, also OPR = sum of the interior opposite angles 08P, 80Py FOCAL PKOPERTIES OF THE ELLIPSE. 35 and QPR = sum of Q8P, SQP, /. sum of OSP, SOP= twice the sum of QSP, SQP, of which OSP= twice QSP, /. angle /S'(9P= twice SQP. Similarly the angle 5' OP' = twice HQP\ And the angle SOP = HOP\ /. SQP= HQF, 14. Any two tangents at right angles to one another intersect on a circle whose centre is C and square on the Let any two tangents at right angles to one another cut the auxiliary circle in Y, Z; F, Z. Draw SY, SY; HZ, HZ. Let QC cut the circle in P, R\ Then SQ, HQ will be rectangles, with opposite sides equal. 3—2 36 EXAMPLES. Hence QR . QE = QT. QZ' = 8Y . HZ=^ BCP, = CA'-vBC\ 15. SG : SP :: SA : AX. The taDgent at P makes equal angles with SF, PH. Hence the normal PG bisects the angle SPIT. Hence (Euclid VI. 3) SG : SP :: EG : HP :: 8G + EG : SP+ PH :: 2SG : 2AG :: SC : AC :: SA : AX, Examples. 1. A series of ellipses pass through a point and have a common focus and their axes major are equal : shew that the other focus always lies on the circumference of a fixed circle. 2. Under the same circumstances the centre also describes a circle. EXAMPLES. 37 3. Under the same circumstances, wLat is tlie greatest excen- tricity tlie ellipse can have, and what does it then become ? 4. Given three points of an ellipse and one focus, shew how to find the corresponding directrix. 5. The lines joining the extremities of a focal chord with the vertices have their points of intersection on the corresponding directrix. 6. The part of a directrix intercepted between the lines joining any point of the ellipse with the vertices subtends a right angle at the corresponding focus. 7. Two circles have their centres fixed and the sum of their radii constant : find the locus of the centre of a cii'cle of constant radius that touches them both. 8. A circle through Z, Z, the feet of the perpendiculars on the tangent from the foci touches the axis major in Q^ and has its centre in the tangent: shew that OQ=BG. 9. If the tangent at a point P, the foot of whose ordinate is at iV, intersects the major axis produced in T -. prove that TX.TG=TA.TA'. 1 0. A circle through F, Z, and N the foot of the ordinate at the point of contact, will also pass through the centre of the ellipse. 11. The focal distances of two points wtere the tangents are parallel form a parallelogram. 12. SY is perpendicular at the tangent at P, HY parallel to PS meets TS produced in F, prove HY' - lAG, 13. If (7P, perpenciicular to the tangent at P intersect IIP in F,HF=AC. 14. The locus of the intersections of the perpendicular from 38 EXAMPLES. the centre on the tangent with the focal distances is two circles with radius AG, and centres at the foci. 15. If CD cut SF and HP in E and E\ prove SE=HE' and the X5ircle round /S^<7=that round HEV, 16. If from the centre of an ellipse lines be drawn parallel and perpendicular to a tangent at any point, they inclose a part of one of the focal distances of that point which equals the other focal distance. 17. li SP, HP are at right angles to one another SP.HP=2BC\ 18. Given the two foci and one tangent of an ellipse to draw the directrices. 19. If from any point ^ of a tangent at P, TQ be drawn at right angles to SP produced if necessary and TRN at right angles to the axis major cuts the curve in R, then SQ = SP. 20. The tangent at the extremity of the latus rectum meets the axis major at the foot of the directrix. 21. The circle that passes through L, L' the extremities of the latus rectum through 8 and touches the nearer directrix is touched by LH. 22. Shew that lines drawn from a focus to points on the ellipse at equal distances from the extremities of the axis minor are equally inclined to the tangents at the points. 23. A ship saUs over an elliptic path having its middle point at B, shew that she changes her direction as much as an observer at the focus does the direction of his telescope in watching her. 24. Given a focal chord and the tangents at its extremity, find the second focus. EXAMPLES. 89 25. Given a focal chord and the second focus, find that which lies on the chord. 26. Given PO, QO tangents to an ellipse at the points P, Q ; POQ is less than a right angle and it is known that one focus lies in FQ, find the other and the directrices. 27. If PiV be an ordinate, the angle F]!^T= the angle PSY, 28. If FiV be an ordinate, PiV bisects the angle T:N'Z. 29. If a line through X the foot of the directrix cut the ellipse in P, p, iSF, Sp are equally inclined to either axis. If FSQ be a focal chord FS^ SQ subtend equal angles at X. 30. The circle on FG as diameter cuts SF, HF in K and L, shew that KL is perpendicular to FG and bisected by it. 31. Prove SG : EG :: SY : HZ, 32. SZ and HY each bisect the normal FG, 33. If DR be the ordinate at D, and CD conjugate to CF, the triangles FGN, DRG are similar, and in any ellipse FG is proportional to CD, 34. If QFN be the common ordinate to the auxiliary circle and the ellipse, and the tangents at Q and P meet on the axis major in P, prove TQ i TF w BO : FG, 35. If Oy 0' be the centres of the circles inscribed in the triangle SFH, and escribed on its side 8H, then FO . FO'=CB\ 36. If ET be the side of a parallelogram whose sides touch an ellipse at the extremities of conjugate diameters, and E, T when joined each with the two foci have their joining lines meeting in and 0'; then shew that 0, S, 0', H lie on the same circle; and that the sum of the angles subtended by 8R at E, T and 0'=that it subtends at 0. 37. A diameter CF produced intersects the directrix in F, prove that VS is at right angles to the diameter conjugate to CF. 40 EXAMPLES. 38. Given the centre and directrix of an ellipse, also the directions of a pair of conjugate diameters, determine the position of the foci. 39. If tangents PT, QT meet on the auxiliary circle, prove that SP is parallel to HQ. 40. If a parallelogram circumscribes an ellipse, the lines joining the points of contact also form a parallelogram. 41. A parallelogram is described about an ellipse having two of its corners on directrices, prove that the other two will lie on the auxiliary circle. 42. If a line be drawn through a focus perpendicular to two parallel tangents, the rectangle of its segments made by the focus = ^(7^ 43. A line is drawn through the focus of an ellipse perpen- dicular to a pair of parallel tangents : on this line as diameter a circle is described : prove that the chord of this circle parallel to the tangents such as when produced passes through the other focus = BB\ 44. If a quadrilateral be circumscribed about an ellipse, the angle subtended at either focus by opposite sides are supple- mentary. 45. Circles are described on SPj HP as diameters, and chords of these circles J7, ZJ are drawn at right angles to the axis major, prove that /S'/, RJ produced intersect on the axis minor. 46. The points in which the tangents at the vertices are intersected by any tangent are joined each with a focus; shew that these lines intersect in the normal. 47. The tangents at the extremities of the latera recta on the same side of the axis major intersect on the circumference of the circle through the foci and the points of contact. EXA]VIPLES. 41 48. If tangents PT, QT meet in T and one of tliem QT is produced to any point Q\ prove that the angle PTQ' is a mean between PSQ and PHQ. 49. An endless string of greater length than the circumference of an ellipse which is laid on a sheet of paper is made to pass round it and stretched tight by a pencil : prove that the point of the pencil will trace an ellipse having the same foci as the original. (It may be assumed that the normal to the pencil's path at each point will make equal angles with the dii'ections of the string at the point.) 50. If P be the vertex of a triangle whose base AB is bisected in G. Then if AP . BP+ CP^ is a constant quantity, the locus of P is an ellipse. 51. If ^Z> be the portion of the generating line of a cone which contains the vertex of an ellipse cut from it intercepted between the vertex and a line through G at right angles to the axis of the cone; shew that AD=GS. 52. To cut an ellipse of given axes from a given cone. CHAPTEH V. On the Hyperbola. 1. Definition. Let A VB\ BVA' be the two sheets of a cone, whose axis OVO' is in the plane of the paper and AB A!B the two positions of the generating line in that plane. Let the cone be cut by a plane ABA perpendicular to the plane of the paper which cuts it in the line AA , such that A, A' are points in the gene- rating line of the cone on the same side of the axis VO\ then if this plane cuts the cone in the lines BAp B'A'p', these lines make up a curve which is called an Hyperbola. It is manifest that the two branches of this curve may be prolonged to any length and cannot intersect. Also they are each divided into two equal parts by the line AA' pro- duced : every line in the plane of the section drawn at right angles to A A will meet the curve in two points on opposite sides of AA and at equal distances from it. ON THE HYPERBOLA. 43 AA' is called the transverse axis of the hyperbola, and a line through C the middle point of A A ', at right angles to AA' in the plane of the section and of a magnitude to be specified in the next article, is called its conjugate axis. 2. Every hyperbola may be projected from or into an hyperbola whose axes are equal, called an equilateral or rectangular h3rperbola. Let DED\ RPR' be circular sections of the cone, made by planes at right angles to the axis through G and through N any point of AA' produced, and let this latter plane cut the plane of the curve in PNF at right angles to AA\ Then NP':=RN.NR\ 44 CONIC SECTIONS. Also in the similar triangles NAR, CAD RN : DC :: AN : AC, and in the similar triangles NA' R\ CA' D' NR : CD' v.A'N-.AlG iiA'NiAC /.RN.NR' : CD. CD' :: AN.A'N:AC\ or NF' : AN. A'N :: CD . CD' : A C. Now if we draw CE from C to touch the circle DED in E, CE^— CD . CD' : and we may now complete our definition of the conjugate axis of the hyperbola by saying that it is equal in length to CE. Now let us take a cone whose vertical angle is a right angle, and let an hyperbola be cut from it by a plane parallel to the axis and such that the transverse axis shall ON THE HYPEEBOLA. 45 equal AA' , then B, B' will merge in F, and CE will = CF", will also = GAy since the circle with centre G and distance GA or GA! will pass through F^ A VA being a right angle. Hence the hyperbola will be rectangular and will have each of its axes equal to AA. Let the two hyperbolas be now placed in the same plane with their transverse axes coincident, and let the ordinate NF of the first hyperbola produced if necessary meet the rectangular hyperbola in Q, Then we have NF" : AN. AN:: GE" : A G\ and NQ' : AN: AN:: AG': A G\ or NQ' = AN. AN: hence NF' : NQ' ::GE':AG, or NF:NQ::GE :AG. We shall shew that GE may be greater or less than GA : if GE be greater than GA the first hyperbola will project into the rectangular one if, the transverse axes remaining coin- cident, they are placed in planes inclined to one another at the proper angle. If GE be less than GA the rectangular hyperbola will project into the one of unequal axes. Cor. The proposition proved above NF':AN.AN::GE':AG'' might have been equally proved if we had taken the circular section in the other sheet of the cone below A. Hence we observe that if we take two points in the transverse axis of the curve produced both ways on opposite sides of G and equi- distant from it, the ordinates drawn through them will be equal, and the points in which they meet the curve will be equidistant from the centre. Hence the two branches of the 46 CONIC SECTIONS. curve are equal and every line through the centre joining them is bisected in the centre. 8. It has been shewn that when the cone has a right angle at the vertex, and the cutting plane is parallel to its axis, the two axes of the curve are equal. Also if the cutting plane is parallel to the axis of the cone, CE— CA > or < GA according as angle of the cone is obtuse or acute. Hence it appears that the axes of an hyperbola may be equal or either in excess of the other. It may be shewn that when the angle of the cone is acute the transverse axis is the greater for all sections ; when the angle is a right angle the axes are equal when the cutting plane is parallel to the axis of the cone, and the transverse axis the gi-eater for all other sections. When the angle of the cone is obtuse, either axis may be greater than the other, or they may be equal according to the inclination of the cutting plane to the axis of the cone. This will appear if we draw VE at right angles to the axis of the cone to meet AA' in E and turn A A' about E, observing that CD. CD' :AC':: VE' : AE.EA', and that AE. EA! is least when A A' is at right angles to jB/Fand diminishes the more the further it is turned from that position. Further, it is evident that hyperbolas cut from any cone will have their axes in the same proportions when the cutting plane is so situated that CD . CD' : A A' is a constant ratio. This will be the case when the plane moves parallel to itself: as in that case each of the ratios CD : AC and CD ''. A' C remains constant. If the angle of the cone is made to vary as well as the position of the cutting plane, a curve equal in all respects will be obtained if AA' remains unchanged and the rectangle CD . CD' is also constant. ON THE HYPERBOLA. 47 We may now proceed to investigate the properties of the rectangular hyperbola, and then we can generalise them by projection for the hyperbola of unequal axes. 4. Property of the Asymptotes. Let PAF be one branch of the curve, and through C draw CR, CE each making half a right angle with CA : let RPNFR be at right angles to A' A produced. Then NR = I^R' = CJSF, and since A' A is bisected in C, and pro- duced to iV", but or .-. AN.A'N+AC'=CN': AN.A'N=NP\ :. NP' + AC^^CN', NF' + AC\=IfR\ 48 CONIC SECTIONS. But RR' is divided into two equal parts in N and un- equally in P : .'. Fr+RP.PR'^NR', .'. RP,PE = ACr, Now the further N is removed from A, the greater does RR' become, and therefore the greater does PR' and the less does PR become, and by increasing AN and therefore PR' we may make PR smaller than any assigned quantity; hence the curve approaches nearer and nearer to the straight line without ever coinciding with it; hence the straight line is said to touch the curve at an infinite distance, and is called an asymptote. CR and CR' both approach the curve in the same way, and each approaches the two branches at its opposite extremities. They are both asymptotes and the rectangular hyperbola is so called from the fact, that its asymptotes are at right angles to each other. 5. The intercepts on any straight line between the rectangular hyperbola and its asymptotes are equal. In the last Art. since iVP = iVP' and -^72 = iTO', there- fore PR = P'P', this proves the proposition for lines at right angles to the transverse axis. Let any chord Qq be produced to meet the asymptotes in U and u. Let the ordinates at Q, q, meet the asymptotes in R, R\ r, r'. Then QU : QR :: qU : qr, Qu : QR' :: qu : qr . Hence QU.Qu : QR. QR' :: qU.qu : qr.qr, ON THE HYPERBOLA. therefore QU, Qu : AC :: qU,qu : A(P, therefore QU. Qu = qU. qiu 4d Let Fbe the middle point of U% then QU, Qu + VQ" =VIP=^ Vu' = qU. qu + Vq\ but Q U. Qu ^qU. qu, therefore VQ' = Vq^ ; hence V is also the middle point of Qq, and therefore QU=qu. J. C. s. 4 50 CONIC SECTIONS. Join GVy cutting the curve in P: all cliords of the asymptotes parallel to Qq will be bisected by GV, and therefore also all chords to the curve parallel to the same line. The locus of the middle points of parallel chords is a Straight line through the centre. If we further draw CW parallel to Qq and QTF parallel to VG, and produce QW to meet the other branch of the curve in §', W will be the middle point of QQ\ For the two branches of the curve are in all respects, equal and similarly situated with respect to the asymptotes. Therefore if we draw the chord Q'q parallel to Qq, VG produced will bisect Q'q (in V suppose). ON THE HYPERBOLA. 51 Also Q'V by construction ==WG=QV: hence Q'q will equal Qq, and CV will equal CV: QfCy WV axe parallelograms, and therefore Q'W=rC=VG=QW and W is the bisection of QQ\ And in like manner all chords between the two branches of the curves parallel to QQ' are bisected by the diameter CW, Let QQ' meet the asymptotes in S, s: then since the angle at (7 is a right angle and V is the middle point of Uu, CV= VU, and OF, VU are equally inclined to the asymp- tote ; so .*. are SW, WG being parallel to them : :.SW=^WG=Ws since the angle at (7 is a right angle : W bisects Ss, and since it also bisects QQ\ .*. QS= Q's. Shewing that every straight line that cuts the curve and the asymptotes, each in two points, and the curve in one branch or both, has the intercepts between the curve and the asymptotes equal. Cor. Hence as the chord Uu moves parallel to itself, so that Q and q approach each other and finally coincide in P, the chord ultimately becomes a tangent at P and is bisected in P : shewing that the part of the tangent intercepted between the asymptotes is bisected at the point of contact Each half of the tangent will equal GP. 6. QV^-\-GP^=VlP. Let Ww be the chord to the asymptotes which touches the curve at P. Through P draw rPr the double ordinate to the asymptotes. Then QU: QB ::PW: Pr :: GP : Pr 4—2 52 and CONIC SECTIONS. Qu : QR' :: Pw: Pr' :: CP ', Pr \ :. QU, Qu : QR, QR! :: CP" : Pr,Pr or QU , Qw, AC :: CP' : AC\ .'. QU. Qu = OP and QV^+ CP'^ VU*. 7. The tangents at the extremities of any chord meet in the diameter which bisects it. Let Pp be any chord, Qq a parallel chord adjacent to it, then Vv through their middle points will pass through the centre of the curve. ON THE HYPERBOLA. 53 Join PQ and let it meet the diameter in T. Then VTivTiiFV: Qv::pV:qv, .*. the secant pq also passes through T. Thus at whatever distance Qq may be from Pp, PQ, pq always intersect on the diameter CV. Hence when Qq approaches Pp and finally coincides with it so that PQ, pq, become finally the tangents at P, p, these tangents will meet in the diameter CV» 8. GV. CT^ CP". If QF be an ordinate of any diameter CP, and QT the tangent at Q, then CP is a mean proportional between CI' and OF. Let PE be the tangent at P intersecting QT in E, PE is parallel to QV: draw PO parallel to QT' to meet QV in 0, Join OE; it will be the diagonal of the parallelogram OPEQ and will therefore bisect the chord QP; hence (by the last proposition) EO bisecting the chord PQ and passing through the intersection of the tangents at its extremities is a diameter, and when produced will pass through C, Hence CV: CP :: CO : CE :: CP: CT and CV.CT=CP'. q.e.d. Cor. 1. If P^be the ordinate from Pto the transverse axis, and the tangent at P intersects the axis in jT, we shall have CN. CT= CA\ 54 CONIC SECTIONS. CoE. 2. Hence to draw tangents to an hyperbola from any point jT, join GT and produce it to meet the curve in P; draw the tangent at P by the aid of CoR. 1. Produce CPto a point Fsuch that CFis a third propor- tional to CT, CP: through Fdraw an ordinate QVq parallel to the tangent at P: join TQ, Tq ; these (by the Proposition) will be the tangents required. 9. The Conjugate Hyperbola, Conjugate Diameters. Jf on the conjugate axis of a rectangular hyperbola as transverse axis we draw another rectangular hyperbola, it is called the conjugate hyperbola, and it is manifest that it has the same asymptotes as the first. ON THE HYPERBOLA. 55 Moreover if we draw semi-diameters CP, CD to meet the two curves equally inclined to the asymptotes and therefore to the axes, these will also be equal : and if we join PD it will cut the asymptotes at right angles, in suppose. Take OR on the asymptote equal to GO and join PR, DR. Then PR = CP and therefore touches the hyperbola at P; and DR (= CD) touches the conjugate at D. Also CPRD is a parallelogram. m J2. M Moreover CP produced bisects all chords of the hyperbola parallel to PR or to CD, and similarly CD produced bisects all chords of the conjugate parallel to DR or to CP. Hence CP, CD produced each bisects the chords parallel to the other, and are called conjugate semi-diameters. 56 . CONie SECTIONS. It will also be seen that CD will bisect the chords between the two branches of the curve parallel to P CP, and CP will bisect all chords of the conjugate parallel to DGU, 10. The area of the parallelogram CPRD is constant Draw PN, DM 2A, right angles to the transverse axis: the triangles OPJV, CDM are equal in all respects. For the angles DGM, PCN sire equally in excess and defect of half a right angle, and therefore together make up a right angle, as do also CPNand PCN; hence DCM= CPN and PCN=^ CDM and the triangles are equiangular and equal in all respects ; DM= ON and CM== PJST, But PIP +AC'= CIP, hence also <71P + ^C^= OA^' and PJ\^^ + ^(7' = Z>1P. Now the triangle D(7P= quadrilateral i>JfiVrP together with triangle D CM less the equal triangle CPN = quadri- lateral DMNP. Now produce DP to meet the axis in W, DWM is half a right angle, hence triangles DMW, PNW are half the squares on DM, PN, and ^ P^+ quadrilateral dmnp=\dm\ But PIP-yAC^DM^ or|pi\r*+|^C^ = |2)ir, /. quadrilateral DMNP ==^AG\ Hence parallelogram i)CPJ2 = twice triangle jD(7P= twice quadrilateral DMNP = A C\ Cor. 1. The parallelogram formed by tangents at the extremities of P, P and D, D' of the diameters PCP, DGD' will have the constant area 4lA C^. ON THE HYPERBOLA. 57 Cor. 2. The rectangle PO ,00— half the parallelogram OPRD=:^^AO\ Cor. 3. If PF be the perpendicular from P on the conjugate semi-diameter OD^ we shall have PF , OB— OA^. 11. QV, Yq is proportional to PV, In the rectangular hyperbola, if a double ordinate Qq to the transverse axis cuts PV, a line from any point of the curve parallel to the asymptote OR in F, then we shall have QV. Vq proportional to PV. Join (7P, OV, and we shall have QV. Vq = 4! times the triangle OPV, For Qq is bisected in N, ,'. QV.Vq+VN'^QN", .-. QV.Vq+ YN' + AO' = QN^ -{-AO^^ EJST. 58 , CONIC SECTIONS. but EV.Vr+VN'=^EN\ /. QV.Vq + AG'=RV, Vr. Let VP produced cut Cr in K; draw VM, PL at right angles to CE : then BV : Vr :: MV : Kr :: MV : VK, /. BV, Vr : BV :: MV. VK : MV, or BV. Vr : MV. VK :: BV : MV : but BV^=2MV, MV.VK= twice the triangle CKV; /. BV.Vr = 4i times the triangle CKV, and -40* = twice the rectangle PL.LC = 4* times the triangle CKP, .'. QV. Vq + 4i times the triangle CKP=4i times the triangle CKV, .'. QV.Vq = 4i times the triangle CP V: and since P F is parallel to CB, the triangle CP V is pro- portional to PV: hence also QV, Vqis proportional to PV. 12. Now let us consider what modifications these pro- positions undergo when transferred by projection to the hyperbola of unequal axes. Let us suppose the conjugate axis of any hyperbola less than the transverse. Then we have shewn that the rectangular hyperbola whose axes are equal to the transverse axis will project into it if the planes of the two hyperbolas intersect in AA' and are inclined at the proper angle. Also the asymptotes will project into lines equally inclined to the transverse axis and at such an angle that if DAD' is drawn to them through A at right angles to CA, AD or AD' will =^ BG (the conjugate semi-axis). ON THE HYPERBOLA. 59 Also the conjugate rectangular hyperbola when projected will have BG for its transverse, CA for its conjugate semi- diameter. This is the conjugate hyperbola: both the hyper- bola and its conjugate will continually approach the asymp- totes CD, CD' without ever meeting them. Also FB, P'R, the distances between the curve and asymptotes on the double ordinate will be equal, and the tangents at A, ^'will be at right angles to the transverse axis. All lines perpendicular to the transverse axis will be diminished by projection in the ratio oi AG : BG, hence RP, PB which was equal to A G^ will become BG^, Also V the middle point of any chord Uu of the asymp- totes will also be the middle point of the chord of the curve as in § 5 : and all chords to the curve parallel to Qgi will be bisected by (7FI 60 CONIC SECTIONS. Also the tangent at P where (7Fcuts the curve will be parallel to QFand will be bisected at P: but as the angle WCw is now acute, each half of the tangent is less CP, The proposition of § 6 must be modified by substituting the equal semi-conjugate diameter CD for CP\ CD is parallel to Uu, and hence QVj CD^ FZ7 will be all altered in the same ratio by projection, and we shall have in the projected curve QF* + CD' = QU' or QU.Qu=:CD\ The proof of the properties of §§ 7, 8 applies equally to all hyperbolas, rectangular or not. Referring to § 9 we see that CP, CD conjugate semi- diameters of the rectangular hyperbola will continue to bisect each other's chords after projection : and they are still said to be conjugate to one another. They will no longer be equally inclined to the asymptote, and whilst CM, CN will be un- altered, DM, PJ^ will be diminished in ratio of AC : BC. Hence we shall have generally CM' + CA' = CN' and PIP + CB' = DM", and by addition PN'+ CN'+ CB' = D3P+ CM'+ CA\ or CP' + CB' ^ CD" -v CA\ Also the parallelogram CPRD will be a parallelogram after projection, and its area will be diminished in the ratio oi AC:BC, and :,=AC.BC: and the parallelogram formed by tangents at the extremities of conjugate diameters wm = 4^(7.^a The proposition of § 11 will not be affected by projection, and therefore applies equally to the hyperbola of unequal axes, and it will be extended by a subsequent proposition OK THE HYPERBOLA. 61 (Art. 15.) to the intercepts on a line drawn from any point in the curve parallel to an asymptote made by parallel lines in any direction. 13. If the conjugate axis of an hyperbola is greater than the transverse, we may project it into a rectangular hyperbola having its axes equal to the transverse, and the same results will obtain as in the last Article, except that the angles between the asymptotes that contain the transverse axis will be obtuse instead of acute. 14. Supplemental Chords. If any point Q in an hyperbola be joined with the ex- tremities PP' of any diameter, PQ, P'Q are called sup- plemental chords, and are parallel to conjugate diameters. For if we take the middle point of PQ and join CO, PQ will be parallel to the conjugate diameter to GO, but G is the middle point of PP' and therefore CO is parallel to P'Q J hence PQ, P'Q are parallel to conjugate diameters. 15. The ratio of the rectangles of two intersecting chords of an hyperbola is the same when one or each is moved into any position parallel to its former position. 62 CONIC SECTIONS. Let VP, Vp be the generating lines of the cone through the ex- tremities of one of the chords FOp, being the point of intersection of the two chords. Through V the ver- tex draw VG parallel to FOp of some fixed length, the same for all the chords. Let planes through and C at right angles to the axis of the cone intersect the plane FVCp (the plane of the paper) in the lines B Or, DdC ; and the generating lines VF, Vp in B, r, D, d; then Er, Dd are chords of the circles in which these planes cut the cone. Also Bor is parallel to DdC as well as FOp to VG. Hence by similar triangles FOB, VCB, FO : BO :: VG : GD, and by similar triangles p Or, VGd, Op : Or :: VG : Gd; therefore FO . Op : BO , Or :: VG" : CD. Gd. Now let F'Op' be another chord of the hyperbola with its extremities on different generating lines of the cone, intersecting the former chord in 0, Draw VG' parallel to FOp and of the same length as VG: also B'Or, D'd'G' the intersections of planes through and G' at right angles to the axis with the plane FVG'p. Then B'Or' is a chord of the same circle as BOr was, S:ndB0.0r=B'0.0r, also VG = VG' : and as before FO. Op iB'O.Or' :: VC^ : G'D' . G'd'. ON THE HYPERBOLA. 63 Hence we shall have PO.Op :F0. Op' :: C'D\ C'd' : CD . Cd. Now observe that if the chords Pop, P'op' are moved parallel to themselves into some new position, but so as to be still chords of the hyperbola, the lines F(7, VC will not be affected, but the generating lines on which the extremities of the chords rest will not be the same as before; yet C, U remaining fixed, Bd, D'd' in their new position will be still chords of the same circles as before, and each of the rect- angles CD . Cdy G'D' . Cd' will retain the same value as before. Hence the ratio PO ,0p\ P'O . Op' will be in- variable, and have the same value wherever may be situated, provided the chords are drawn^ always parallel to their original position. The same proof will hold when the extremities of the chords are on different branches of the curve, and when is on the convex side of either branch. The same proof will also hold for the other sections of the cone equally with the hyperbola. 16. The ratio of the rectangles on the segments of the chords equals that of the squares on the parallel semi- diameters. Let the chords move till they become tangents to the curve, viz. the tangents OP, OP intersecting in 0. The ratio of the rectangles is that of the squares on OP, OP. Draw CQ parallel to OP to meet the conjugate in Q, and draw QQ' parallel to PP to meet CQ' parallel to OP in Q'. Join CO, and let it meet QQ' in W, and when pro- duced let it meet PP in V, 64 CONIC SECTIONS. Then V is tlie middle point of PF, and the triangles CWQ, GWQ' are similar to OVF,OVP' : /. QW : WG :: FV : VO, and gw : WG :: FV : VO, ,'. QW : Q'W:: FV : FV, but FV=FV; therefore also QW=-Q[W, But (7F bisecting FF\ bisects all chords of the hyperbola and its conjugate parallel to FFy and therefore QQ', parallel to FF and bisected by GO, is a chord of the conjugate hyperbola; GQ' is the semi-diameter parallel to 0F\ and the ratio of the rectangles under the segments of the chords = OF' : 0F^=^ GQ" : GQ'^ since the triangles OFF, OQQ' are similar. See also Besant's Elementary Gonic Sections, p. 116. Cor. If a circle intersect an hyperbola in four points, it may be easily shewn that the diameters parallel to the ox THE HYPERBOLA. 65 pairs of opposite sides of the quadrilateral formed by drawing the common chords will be equal, and therefore equally in- clined to the axes of the curve ; hence the opposite sides of the quadrilateral will be equally inclined to the axes. 17. NG :NG :: BC : AC\ Let PG be the normal at P; draw NR from the foot of the ordinate PN to touch the auxiliary circle at B ; then if PT be the taogent Sit P, BT will be at right angles to the transverse axis, because CN. CT — CA\ Also NB^^NA . JSfA' and .-. NP' : NB^ :: BC : AC\ and NP : NB :: BG : AC, Join CB : then in the right-angled triangles TPG, CBN, PN" = TN. NG, BN' = CN, TN, .-. PN' : BN' :: NG : JSfC; .'. NG : NC :: NP' : NB' :: BC : AC\ 18. PF.PG^BC, PF,Pg = AC\ If the normal at P intersects the axes in G and ^, and PF be the perpendicular on CD the semi-diameter conjugate to CP, then first, PF.Pg^-A C\ J. c. s. 5 66 CONIC SECTIONS. Let FT be the tangent at P, and let Pnr parallel to transverse axis cut CD and the conjugate axis in n and r. nT, tN will be parallelograms. Then since the angles at F and r are right angles, a circle will pass through F, n, r and g. Hence PF . Pg = Pn . Pr= CT , CN= CA\ Also PG : Pg :: NG : NO :: BC : AC: and PF.Pg = AC'] r.PF.PG = BC\ 19. The asjmaptotes of any hyperbola are parallel to the generating lines of the cone in which a plane through the vertex parallel to the cutting plane cuts the cone. Let VF be the line in which a plane through F parallel to the cutting plane cuts the plane of the paper. Make VE equal to AG and draw FFF' through E at right angles to the axis. EXAMPLES. 67 Then the part of the perpendicular through jS* to the plane of the paper between that plane and the cone will be the height of a rightr-angled triangle which has half the angle between two generating lines at the base, call it EG, Then EG^ = EF . EF': and from equal triangles A CD, VEF and A' CD\ VEF', EF= CD, EF' = CD' ; .-. EG" = CD,CD' = B C and EG = BC, Hence the generating lines are inclined to VE at the same angle as the asymptotes to GA parallel to it, and being in a plane parallel to that containing the asymptotes, are parallel to them. Examples. 1. Op all hyperbolas that can be cut from a given cone, the ratio J3C : AC is greatest in that cut by a plane parallel to the axis. 2. Give a construction for determining, if possible, the direction of the plane that will cut a rectangular hyperbola from a given cone. 5—2 68 EXAMPLES. 3. KPQ is drawn from any point N of the conjugate axis of a rectangular hyperbola at right angles to it, to cut the auxiliary circle and the hyperbola in P and Q ; prove NP^ + NQ^=2AG^. 4. Prove the same for any hyperbola and the ellipse on the same axes. 5. A circle passes through^, A', the vertices of a rectangular hyperbola : the common chord parallel to AA' m a diameter of the circle. 6. The tangent PT to a circle intersects a fixed diameter in Tj and TQ is dravi^n at right angles to the diameter of a length bearing a constant ratio to TP : as P moves on the circle, Q will move on an hyperbola which has the given circle for its auxiliary circle. 7. Through any point P of a circle on AB as diameter PA^ BP are drawn and produced to Q and R such that QR is at right angles to BA and bisected by it : prove that as P moves on the curve Qj R will move on a rectangular hyperbola. 8. Draw a tangent to an hyperbola parallel to a given line. 9. From a given point in an hyperbola draw a line such that the intercept between the other point of intersection and an asymptote shall equal a given line. When does the problem become impossible ? 10. The tangent at P meets an asymptote in T^ and TQ is drawn to the curve parallel to the other asymptote. PQ produced both ways meets the asymptotes in R, R'. RR' is trisected in P, Q. 11. From a given point P in a rectangular hyperbola PJ/, PR are drawn equally inclined to an asymptote, and when pro- duced meet the curve again in Q, R; prove that QR is a diameter. 12. Find the position and magnitude of the axes of an hyperbola which has a given line for asymptote, touches another line in a given point, and passes through another given point. EXA3klPLES. 69 13. Draw lines from the centre of an hyperbola to the extremities of any chord : the intercepts of any line parallel to the chord between these lines and the asymptotes will be equal. 14. From CV. CT=CF^ deduce in the rectangular hyper- bola, VC.VT=Qr'. 15. In the rectangular hyperbola the triangles CVQ^ QVT are similar. 16. In the rectangular hyperbola, E is the middle point of a chord Qq and JiQ\ j-^/ are drawn parallel to the tangents at q, Q to meet CQ and Cq : shew that a circle will circumscribe CQ'Iiq'. 17. From any point i? of an asymptote 7?iV, i^ifare drawn at right angles to the axes intersecting the hyperbola and its conjugate in F and B. Prove CP, CD are conjugate in the rectangular and general hyperbolas. 18. The tangent at P meets an asymptote in T, Tl^is drawn at right angles to the transverse axis ; prove in the rectangular and general hyperbolas that NF passes through D the extremity of the diameter conjugate to CF. 19. Any two tangents have their points of intersection with the asymptotes joined; the lines so drawn will be parallel. 20. From any point P of an hyperbola PiT, FK are drawn each parallel to one asymptote to meet the other : these lines pro- duced if necessary meet any line through the centre in P and T. Complete the parallelogram FRQT, and shew that ^ is a point on the curve. 21. Draw a tangent to an hyperbola at a given distance (less than AC) from C. 22. The tangent to an hyperbola meets a pair of conjugate diameters in T, t and the second tangents are drawn to the curve from T and t^ they will touch it at the extremity of a diameter. 70 EXAMPI^ES. 23. An hyperbola can be drawn througb the extremities of any two radii of a circle having the diameters at right angles to the radii as asymptotes. 24. An hyperbola can be drawn through the extremities of any two semi-diameters of an ellipse having the diameters conju- gate to them as asymptotes. 25. If PG be the normal at P, GG = WF in the rectangular hyperbola. 26. PG.Pg = CD\ 27. If the tangent at P intersects the asymptotes in i?, r the circle on Gg as diameter will pass through C, P and r. 28. In the same hyperbola Gg varies inversely as the per- pendicular from the centre on the tangent. 29. An hyperbola is cut from a given cone, and a straight line drawn from a point of it parallel to an asymptote; the plane through the vertex of the cone and this line will cut the cone in two straight lines one of which is parallel to the line in the curve. Hence prove the proposition of Art. 1 1. 30. Apply the method of proof in Art. 15 to shew that if parallel tangents at ft Q' meet the tangent at P in Ty T\ QT : PT :: Q'T ; P'T\ Prove the following propositions in the rectangular hyperbola : 31. The lines bisecting the angles between CP and the tangent at P are parallel to the asymptotes. 32. The tangent at the point Q intersects a pair of conjugate diameters in Ty T': prove that CQ is the tangent to the circle round CTT. EXAMPLES. 71 33. If F be the middle point of a chord, the lines bisect- ing the angles between CV and the chord are parallel to the asymptotes. 34. The lines bisecting the angles between supplemental chords are parallel to the asymptotes. 35. The angle between lines drawn from two points on the curve to one extremity of a diameter equals or is the supplement of that between the lines from the same points to the other extremity. 36. Diameters at right angles are equal to one another. 37. Of two chords of the curve at right angles to one another one has its extremities on the same branch, the other on different branches. 38. If AB, CD are chords at right angles to one another and the circle ABC cuts CD produced in Uj BA will bisect ED. 39. The four circles that may be drawn through three of the points AyB, Cf Din the last example consist of two pairs of equal circles. 40. If a tangent be at right angles to a chord, the circle on the chord as diameter will pass through the point of contact. 41. The line joining one end of a diameter with one end of a chord at right angles to it is at right angles to the line joining the other end of the diameter with the other end of the chord. CHAPTER VII. On the Focal properties of the Hyperbola. 1. Like the Ellipse, the Hyperbola has two foci and directrices: before treating of them let us premise the following propositions: If the circles escribed on the sides VA, VA of the triangle VAA touch them and the produced sides in X, S, K, L\ H, K respectively, then AS = AE and KH or K'L =^AA\ ON THE FOCAL PROPERTIES OF THE HYPERBOLA. 73 For SH==: SA + AH=AL-{- AK' = 2AL + LK\ bIso =SA' + A' H=A' K+A' r^2A' L' + Kr: and LK' = KL\ .'. AL = A'L' otAS= A'H. Add AA' to each of these equals, ,:AS+AA' = AH=AK' = AL + LK' = AS + LIC; :,AA' = LK'otKL\ Also, if we produce KL, K'L' to meet A A' in X, X' (as in the figure of the next article), AX— A' X, For since AL = A' L' and these lines are equally inclined to the line joining the centre of the circles, their projections on this line will be equal : and AX, A' X' have the same projections on the same line as AL, A' L' and therefore have their projections on it equal, and being in the same straight line are themselves equal. 2. Now let AA' be the line in which a plane perpen- dicular to the plane of the paper cuts it. Let AP, ^' P* be the two branches of the hyperbola in which the same plane cuts the two sheets of the cone A VK, A' VK' whose axis is the line joining the centres of the circles in the last Article : and let the circles SLK, HL' K revolve about the axis, they will form spheres that touch the cone in circles LRK, LICK'- Let P be any point in the hyperbola: the generating line P F of the cone will touch the spheres, in points P, B! suppose. Join /SP, J3P; these lines lie in the plane that touches the two spheres at B and H, and therefore touch them in those points: hence SP = PB, HP=PR'y 74 CONIC SECTIONS. .-. HP- SP= PR -PR=^ BR' = KL' = A A', a distance independent of the position of the point P : hence the differ- ence of the distances of any point in the hyperbola from S and H is constant. S and H are called the focL 3. Now let the plane through P perpendicular to the axis of the cone cut the cone in the circle QPQ' and the plane of the hyperbola in NP at right angles to A' A produced. Then we shall have SP=PIl=QKf and the triangles QAN, KAX are similar ; /. qK'.NX::AK'.AXv.A8:AX. Hence, wherever P is situated on the curve SP : NX is a constant ratio ==AS: AX, ON THE LOCAL PROPERTIES OF THE HYPERBOLA. Similarly, joining HPy HP=^PR'=QK\ and the triangles QAN, K'AX' are similar ; .-. OK' : NX' :: AK : AX' :: AK: AX:: AS : AX. 75 Hence also HP : NX^ is a constant ratio the same as SP : NX, wherever P is situated on the curve. We have shewn that HA' : A' X' = 8 A : AX : these ratios = A'L' : A'X' or AK: AX; and the triangles AXK, A' X' L' are both right angled at X or one of them has an obtuse angle at X\ in either case we see that the ratios are ratios of greater inequality. As in the ellipse either of these ratios is called the eccentricity. 76 CONIC SECTIONS. So then if we draw the hyperbolic section in the plane of the paper and draw through X, X lines XZ, X Z' at right angles to XX' and PM, PM' perpendiculars to these lines, we have for any point P of the hyperbola 8P:NX (or PM) HP'.NX {oxPM') ratios of greater inequality. 4. The position of the foci is determined relatively to the magnitude of the axes by the following relation CS^ = CA' + GB\ We have shewn that by giving the proper angle to the cone any hyperbola of given axes may be cut from it by a plane parallel to the axis. Let then the given hyperbola be made by the intersection of a plane parallel to the axis of the cone. The foci will still be the points where the inscribed spheres touch the ciftting plane, and these spheres ON THE FOCAL PROPERTIES OF THE HYPERBOLA. 77 will be equal. The figure represents the section of the cone and spheres by the plane of the paper. CV will now be perpendicular to the axis and equal to CB. Also AA' = KK' = 2irF since the circles are equal : r.CA^KV. Hence CS=^ CA-\- AS=^ KV + AK= AV, .-. CS' = AV'= CA'+ Cr= CA'-^CB\ 5. CX is a third proportional to C8 and CA. For in the last figure KX is parallel to OF, /. AV: AG :: KV : XC, or CS-: CA :: CA : CX. Cor. Each of these ratios == AK : AX or AS: AX, the eccentricity. 78 CONIC SECTIONS. 6. Properties of the secant and tangent. Art. 6. of Chap, Y. applies verbatim to all the Conic Sections. 7. The focal distances make equal angles with the tangent at any point. Let the tangent at P meet the directrices in F, F, join 8F, EF'\ F8F, PHF' will be right angles. Also the triangles MPF, M' PF' are similar : and SP'.FMi: AS : AX :: HP : PM\ ,\ SP : PR:: PM : PM' -a PF'.PF\ or BP : PF :: HP : PF'. Hence (Euclid YI. A.) SPF, HPF are similar triangles, and the angles SPF, HPF' are equal CoE. The tangents at A, A' the vertices of the curve are at right angles to the transverse axes. ON THE FOCAL PROPERTIES OF THE HYPERBOLA. 79 8. The feet of the perpendiculars from the foci on any tangent lie on the circle on AA' as diameter. Let SYy the perpendicular from 8 on the tangent at P, meet HP in K: join SP, GY. Then in the right-angled triangles 8YP, KYP, PY is common and the angle SPY = the angle HP Y (by the last proposition). Hence 8Y= KY, and 8P= KP ; ,\ KH=HP^ KP= HP- SP = 2Aa And C Y being the middle points of SH, SK, CY is parallel to HK and half of it and therefore =AC. Hence Y lies on the circle on A A' as diameter. And similarly Z the foot of the perpendicular from H, Cor 1. GY being parallel to HP and bisecting SH^ when produced will also bisect SP. Hence SYP being a right angle, the circle on SP as diameter passes through Y and has its centre on OF produced ; hence it touches the auxiliary circle at F, Cor. 2. If GHhe drawn parallel to the tangent at P 80 CONIC SECTIONS. and therefore conjugate to CP and intersect PH'vn E, then GEPYi^ a parallelogram and P^= CY=AC. 9. If SY, HZ are perpendiculars on the tangent at P from the foci S, H, SY, HZ= BC\ Since (in the figure of the last Article) YZH is a right angle, YG produced will meet ZH on the circumference of the circle. The triangles SGY, HOY' are equal in all respects, and SY.HZ=^HY'.HZ=HA.HA' = HG'-A'G'=^BG\ 10. SP.HP=GB\ Art. 10 of Chap. V. applies verbatim to the hyperbola. 11. If from any point Q of the tangent PK perpendi- culars QN', QT be drawn to the directrix and SPy then ST : QK :: AS : AX. 12. Hence if QP, QP be the tangents to an hyperbola from an exterior point Q, QP, QP subtend equal angles at S, See Articles 11, 12 of Chap. Y. . If P, P' are not on the same branch of the curve, since Q lies on the near side of the directrix to one branch and the further side to the other, T or T will lie in PS or P'S pro- duced, and the supplement of one of the angles QSP, QSP[ will be equal to the other, i.e. the angles will be supple- mentary. Cor. Combining this last proposition with that of Art. 7 we see that the point Q is equidistant from the four lines SP, HP, SF, HF. 13. If QP, QP* be the tangents to an hyperbola from a ON THE FOCAL PROPERTIES OF THE HYPERBOLA. 81 point Q, the angles SQP, HQP' are supplementary, as are also SQP\ HQP. .1" Join SP, SQ, SP\ IIP, HQ, HP', The angles of the quadrilateral 8PIIP' are bisected by the lines drawn from the angular points to Q ; .-. the sum of the angles SPQ, PSQ, HP' Q, P'HQ=- sum of P'SQ, SP'Q, PHQ, HPQ. But the first sum of angles with SQPy HQP' = the sum of the interior angles of two triangles = the second sum with SQP, HQP: hence SQP, HQP' together =^ SQP, HQP together : and these four angles together = four right angles : hence /S'QP + ^(gP' =^§P' + ^QP= two right angles. Q.E.D. If P, P' lie on . different branches it may be shewn, as in the case of the Ellipse, that SQP, HQP' are each half of the angle between HP and SP', and therefore (no longer supplementary but) equal to one another. 14. Any two tangents at right angles to one another intersect on a circle whose centre is C and square on the radius + 50' = -4 (7'. j.c.s. 6 82 CONIC SECTIONS. Let any two tangents at right angles to one another in- tersect in Q and cut the circle on A A' as diameter in Y, Z; r, Z', Draw SY, SY'; HZ, HZ. Let QC cut the circle in B, F. Then SQ, HQ will be rectangles with opposite sides equal. Hence QR . QR'=:^QY . QZ =^ 8Y . HZ=BC\ and CQ'+QR.QE^CR'; ,'. CQ' + BG'--AC': and the locus of Q is a circle with centre G and square on its radius = difference of the squares on AC and BC, Cor. If A Che less than BO the curve has no tangents at right angles to one another : but in this case the circle will be the locus of the intersections of tangents to the con- jugate hyperbola at right angles to one another. ON THE FOCAL PROPEKTIES OF THE HYPERBOLA. 83 15. SG : SP :: 8A : AX. The tangent at P bisects the angle SPII: hence the normal PG bisects the angle between SP and ITP produc3cL Hence (Euclid vi. A) SG : SP :: EG : HP :: HG-SG : HP - SP :: 2SG : 2AG :: CS : CA :: SA : AX. Examples. 1. Given a focus and two points of an ellipse, the other focus lies always on a certain hyperbola. 2. Four circles are drawn having the comers of a square for centres and diameters equal to the side : the locus of the centres of all circles which touch opposite pairs of circles, one internally and the other externally, is a pair of rectangular hyperbolas which have the common tangents of the circles for asymptotes. 3. The tangent from C to the circle through the foot of the directrix and the extremities of the latus rectum equals CS. 4. If N be the middle point of SX, prove AX . A'X=SN\ 6—2 84 EXAMPLES. 5. The portion of either asymptote between the directrices equals the transverse axis. 6. Given the asymptotes and directrices, find the foci. 7. Given one asymptote and the direction of the second and a focus, find the vertices. 8. Given the asymptotes and one point on the curve, find the foci. 9. The focal distance of any point P on the hyperbola equals a line drawn from P parallel to an asymptote to meet the cor- responding directrix. 10. Each of the tangents drawn to the auxiliary circle from the foci equals BG and touches it in a point where the directrix cuts it. 11. Two hyperbolas have the same asymptotes, shew that the chord of one touching the other is bisected at the point of contact. 12. Two hyperbolas have their foci coincident and the angles between the asymptotes supplementary : no tangent to one can be at right angles to a tangent to the other except the asymptotes. 13. The line drawn from aS' parallel to an asymptote to meet the curve equals a quarter of the latus rectum. 14. Two chords through the same focus have three of their four extremities on one branch, and the remaining extremity on the other : prove that the four lines joining their extremities inter- sect two and two on the corresponding directrix. 15. FQ a chord of one branch of an hyperbola subtends at aS' an angle double of that subtended at aS' by a length 'pq on the cor- responding directrix : shew that Pp, Qq intersect on the curve. 16. An ellipse cuts any confocal hyperbola at right angles. EXA^IPLES. 85 17. A circle is drawn through the foci and any point P of an hyperbola, the tangent and normal at P meet the conjugate axis in the same points as this circle. 18. Py Q are points in two confocal hyperbolas at which SH subtends equal angles, the tangents at P, Q are inclined at an angle equal to that subtended by PQ at either focus. 19. HF' drawn parallel to SP meets SY produced on the circumference of a fixed circle. 20. Given one focus, one point and one tangent of an hyper- bola, the locus of the other focus is an hyperbola. 21. Qq Si chord of the asymptotes moves parallel to itself and tangents are drawn from Q, q, their intersection will lie on a straight line through C. 22. A rectangular hyperbola confocal with an ellipse cuts it at the extremities of equal conjugate diameters of the ellipse. 23. An ellipse and confocal hyperbola intersect in P: one asymptote passes through the poiat of the auxiliary circle of the ellipse corresponding to P. 24. An ellipse and hyperbola have each their foci at the ver- tices of the other : if the tangents at the point of intersection meet the conjugate axis in t, t', Ct= Ct' . 25. The tangent at P is perpendicular to one asymptote and Pas' through the focus meets that asymptote in Q^ prove SQ = AC. 26. P is the point of the hyperbola where SP is at right angles to HP : and they intersect CD in £, E'. Prove EE"" = 2 AC and CD' = 2BC\ 27. From the point of intersection of an asymptote and directrix a tangent is drawn to the hyperbola : the line joining the interior focus of the branch it touches with the point of contact is parallel to the asymptote. 86 EXAMPLES. 28. The tangent from P a point in the asymptote touches the curve in : HT is parallel to the same asymptote; prove that HP bisects the angle THO : and if PO intersects the other asymptote in Q, PHQ = hsiU the angle PCQ. 29. The focal distances of two points P, P' intersect in 0, ])rove that the tangents QP^ QP subtend equal angles at 0. 30. If SP, HQ are parallel, find the locus of the intersection of the tangents at P and Q. 31. PT, QT tangents from T b, point in the auxiliary circle whether to the same branch or not have one of the focal distances SP, SQ parallel to one of the two HQ, HP 32. The tangent at P is perpendicular to two parallel tangents at Q, Q\ prove that &Q, HQ' subtend equal angles at P. 33. \iGL be the perpendicular from G on SP, prove GL : PN a constant ratio. 34. From the vertex A draw ^^ perpendicular to the tangent at P, and let QA produced meet Pas' produced in 0, the locus of is a circle. 35. Let G Fproduced meet ^^ produced in Z', then HZ'= HZ. 36. • If the normals at P, Q extremities of a focal chord inter- sect in 0, and OL parallel to the transverse axis cut SP in L, L is the middle point of PQ. 37. If GR be the perpendicular on SP from G, PR equals semi-latus rectum. 38. If the normal at P meets the axes in G, g, the triangles SPg, GSg are similar, and Gg : Sg and therefore Pg : Sg are constant ratios. CHAPTER VIII. On the Parabola. 1. We have now to consider the ease in which the cutting plane is parallel to one of the generating lines of the cone. Let the cutting plane intersect the cone in the curve PAp, and intersect the plane of the paper which contains the axis in the line AN parallel to the generat- ing line VB in the plane of the paper, then PAp is called a Parabola. 8S CONIC SECTIONS. It is manifest that the parabola is divided into equal parts by the line AN" which is called the axis, every line in the plane of the section at right angles to AN" will meet the curve in two points on opposite sides of A]^ and at equal distances from it. The axis intersects the curve in one point only, in other words the curve, has only one vertex. 2. By turning the catting plane about a line through A at right angles to the plane of the paper through the smallest possible angle, the parabola is changed either into an ellipse or hyperbola whose centre is at a great distance from C: the parabola may therefore be considered as the form to which the semi-ellipse cut off by the axis minor on the side of A and one branch of the hyperbola approach more and more nearly when, the vertex remaining fixed, the centre is removed to a greater and greater distance. Hence it appears that lines drawn from all points in the semi- ellipse and semi-hyperbola to their centres become more and more nearly parallel as the cutting plane moves towards the position in which it cuts a parabola from the cone. And we may anticipate that the line joining the middle points of parallel chords of a parabola will be parallel to its axis; and generally all properties of the ellipse and hyperbola that relate to lines that remain finite in the parabola are equally true in that curve. It will have been observed that there is a similarity in the properties of the ellipse and hyperbola ; the parabola is the curve in which they meet, or in which each undergoes the transition into the other. ON THE PARABOLA. 89 Let RPR' be a circular section of the cone made by a plane at right angles to the axis through N, any point of the axis of the parabola and cutting the plane of the curve in PNF at right angles to the axis. Then NF' = RN.NR',8ind the ratio NR : AN is constant for all positions of iV. Hence the ratio NF^ : AN . NR' is constant : and NR^ is also invariable. The ratio RN: AN is invariable for all positions of iA'; the ratio NR : AN may be made to have all values from a ratio indefinitely great to one indefinitely small by moving N along the axis from A. Hence we may find a point S in the axis such that for that point RN . NR' = 4iAN^, or if LSL' be the double ordinate through >S', SD = ^AS^ or 90 CONIC SECTIONS. Hence generally NP' : 8L' :: AN. NE' : AS. NE' :: AN : AS, NF' : 4^AS' :: AN : AS, .'. NF' = 4^AS . AN. 4. QE. Q'E^i^AS.PR. Let FR from any point F of the curve meet the double ordinate QMQ' in R, then QR .Q'E = 4^AS. FE. Draw the double ordinate FNF\ then PM is a rectangle whose opposite sides are equal: and since QQ' is bisected in M, QE . QE + EM' = QliP, or Q^ . gi2 + FN"" = Qil/'; .-. QR . Q'E + 4>AS . AN = ^^/S'^if ; .-. QR . Q'E = 4^/S^ . MN = 4^/S' . FE. ox THE PARABOLA. 91 This proposition is equally true when, as in fig. § 6, ^ is a point in the chord QQ' external to the curve. This property is seen in the case of the hyperbola in Chap. V. § 11. By Chap. vi. § 19 it appears that as the plane that cuts an hyperbola from the cone moves about the line through A towards the position in which it cuts a parabola, the angle between the asymptotes diminishes and they be- come more and more nearly parallel to the axis. Hence, as the hyperbola passes into the parabola, the line from any point of the curve parallel to an asymptote becomes parallel to the axis : and the proposition of the present Article takes the place of that previously proved for the hyperbola. 5. The middle points of parallel chords lie on a straight line parallel to the axis. Let V be the middle point of any chord PQ, draw the ordinates PF, FO, and the double ordinate QMQ', PE parallel to the axis. 92 CONIC SECTIONS. Then since Q Q is bisected in if, nq = BM^ MQ = FN+ QM= 2V0, and QR , BQ = 4^AS . PR or QR . VO = 2AS . PR, and VO : 2AS :: PR : QR a constant ratio for all parallel chords. Hence VO is invariable for parallel chords, and all their middle points lie on a straight line parallel to the axis. This line is called the diameter of the chords. This property corresponds to the fact that the ellipse and hyperbola have the middle points of all parallel chords on a straight line through the centre. 6. RQ' = 4^AS.PV, From V the middle point of any chord Qq draw VP parallel to the ^xis to meet the curve in P. Let P V produced both ways meet the double ordinates through Q, q (the former produced) in ^, r: F will be the middle point of Rr, and Pr = PV + Vr = PV ^- VR = PR + 2PV ON THE PARABOLA. Also qr = QB, and qr = Q' R + 2Qi?; /. ^r . qr = Q^ . Q R + 2QJ?^ or 4^/Sf.Pr = 4^/S'.Pi^ + 2Q22^ QR' + 2AS.PR = 2A8, Pr = 2AS.PR + 4^AS.Pr; 7. Tlie point /S^ we shall see is the focus of the curve, and we may now conveniently consider the properties of the curve with relation to it. We premise the following propo- sition. If a circle touch the parallel lines VL, AS, and the line AV that intersects them in L, 8 and K, then if LK produced meets SA produced in X, AS === AX. For VK= VL, .\ the angle VLK=VKL: but VLK = alternate angle AXK, and VKL = vertical opposite angle AKX: .-. AXK=AKX and AK=^AX, but AS=^AK', .'.AS^AX, 8. Now let AN be the intersection of the plane of the paper with a plane at right angles to it that cuts the cone 94 CONIC SECTIONS. QVQ' in the parabola PAP', so that AN is parallel to VQ' P' ^---^"^"^/V "^"^^^^ V^^^— -CZ\ — — — — -^' Ir- ^"V^ a' Describe a circle SKL in the plane of the paper touching the generating lines VQ, VQ in K and L, and AN in S, Then if we make the circle revolve about its diameter which coincides with the axis, it will generate a sphere which will touch the cone in the circle KRL. Every point of this circle will be equidistant from F. Let a circular section QPQ' through N cut the parabola in P and the plane of the paper in QNQ' : the triangle QAN is similar to Q VQ' : /. NA=QA: and AX^ AK, .\ NX= QK Join SP ; also PV cutting the circle KRL in B. PR will =i QK. And SP, PR tangents drawn to the sphere SKL from the point P are equal : r. SP = PR = QK— NX : hence the distance of any point of the curve from S^ the distance of the foot of its ordinate from X ON THE PARABOLA. 95 So then if we draw the parabolic section in the plane of the paper and draw through X a line XM at right angles to the axis, and PM at right angles to XM, then we have for any point of the curve SF=^NX=PM: the distance of any point F from >S' = its perpendicular distance from XM. As before, S is the focus and XM the directrix of the parabola. The eccentricity of the parabola is thus seen to be a ratio of equality. This results also from the fact that the para- bola is the limiting form of a semi-ellipse, or of one branch of the hyperbola, when the centre moves to a continually increasing distance from the vertex, whilst the focus ap- proaches a position at a certain definite distance from the vertex. In this case AS: AX i^nd^ to become a ratio of equality. For in the ellipse and hyperbola C8. CX— GA^) therefore if C8, CX are the distances of an external point C from the points where a line from G through the centre of a circle cuts the circle, GA will be the length of the tangent : and as G moves to a continually increasing distance, the tangent will become more and more nearly parallel to the diameter through (7, and GA will tend to become an arithmetic mean between CS, GX. Hence as the semi-ellipse or branch of the hyperbola passes into the parabola, the vertex will assume the position midway between S and X, and the eccentricity will become a ratio of equality. 9. The position of S is identical with that previously assigned to it in proving the relation FN^ *» 4iAS, AN, For 96 CONIC SECTIONS. SL the semi-latus rectum or ordinate through S will — 8X = 28A, which is the relation by which S was pre- viously determined. 10. Properties of the secant and tangent. Art. .6 of Chap. IV. applies verbatim to the Parabola. 11. The tangent at any point makes equal angles with the focal distance of the point and the axis of the curve. Let the tangent at P meet the directrix in F and the axis in T: join SP, SF, and draw the perpendicular PM. By the last Art. PSF is a right angle, also SP= PM; hence the right-angled triangles PSF, PMF having equal heights and hypotenuse common are equal in all respects : the angle SPF=MPF=PT8. GOR. 1. Hence 8P= 8T, and the perpendicular on the tangent from 8 will bisect the angle P8T, Cor. 2. The angle between the focal distance and the tangent at any point cannot be a right angle except at the vertex, when 8P, PM are in the same straight line and the tangent makes equal angles with them. ox THE PAEABOLA. 97 12. Tangents at the extremities of a focal chord inter- sect on the directrix in a right angle. Let PF, QF, tangents at the extremities of the focal chord PSQ, intersect on the directrix in the point F and cut the axis in points 2] T. Then the angle PFQ = FTT' + FT'T =^PTS+QrS ==QPF-\-PQF = half the sum of the interior angles of the triangle FPQ = a right angle. 13. The foot of the perpendicular from the focus on any tangent, lies on the tangent at the vertex. Let PR the tangent at P intersect BM in R. Then in the triangles SPR, MPR, SP=PM, and the angle SPR = MPR, hence the triangles are equal in all respects and the angle SRP = MRP, SM is at right angles to PR. Join AR ; then SX is bisected in A and SM in R : .'. AR is parallel to XM and is perpendicular to the axis and touches the parabola at A the vertex. Hence the foot of the perpendicular on PR from S lies on the tangent at A. Cor. If be the middle point of SP, RO will be parallel to 3IP, and therefore perpendicular to AR, and the circle on SP as diameter will touch AR at R.AR is the limiting form of the auxiliary circle in the ellipse and J. C. s. 7 M ■Py > / N. X J^ L S 98 CONIC SECTIONS. liyperbola when the centre removes to a continually increas- ing distance. 14. 8T: QN::AS:AX, as in Chap. IV. § 11. 15. The tangents to a parabola from an external point subtend equal angles at the focus, as in Chap. IV. § 12. 16. The angle between two tangents to a parabola equals that subtended by either tangent at the focus. Let SY, SZ be the perpendiculars from the focus on the tangents to the parabola at F and Q which intersect in 0. Join 8P, SQ, 80. Then the angle A8Y= half A8P, and ^>^^= half A8Q, Also the angle between the tan- gents equals the angle between the jDerpendiculars on them ^Z8Y=A8Z-A8Y =iA8Q-iA8P=iP8Q = 08PoT 08Q, 17. The triangles 80P, 8Q0 are similar and 80' = 8P.8Q. The same construction being made as in the last proposi- tion, we shall have the sum of the angles 8 OP, POZ= 80Z = the sum of the interior and opposite angles of the triangle 8Q0 = i}iQ sum of 8Q0, 08Q. But P0Z=08Q: /. 80P= 8Q0. And the angle P80 = 08Q. Hence the triangles 80P, 8Q0 are similar and 8P: 80 :: 80 : 8Q or 80' = 8P. 8Q. Cor. If /S^Fbe the perpendicular from 8 on the tangent at P, YA will be the tangent at A, and 8Y'=-8A . 8P. ON THE PARABOLA. 99 by 18. The circle circumscribing the triangle formed three tangents to a parabola passes through the focus. Let two of the tangents intersect in and the third tangent cut them in P, Q and touch the curve in E. Join SR and produce QO to Y. Then the angle OPQ = PSR, and the angle OQP=QSR: hence the angle POY= OPQ-\- QP = PSR+ QSR=QSP: the sum of the opposite angles POQ, PSQ of the quadri- lateral OPSQ = the sum oi POQ, POY= two right angles, and the quadrilateral can be inscribed in a circle. In other words, the circle circumscribing the triangle POQ will pass through S. 19. The tansrents at the extremities of a chord intersect on the diameter of the chord. 20. PT=PV. KSee Chap. v. § 7. Let QT' be the tangent at the point Q and QV the ordinate at that point to the diameter PV'. to shew that PT= PV. For Q F will be parallel to PR the tangent at P. Draw PO parallel to QT to meet QFin 0. Join RO'. being 7-2 100 CONIC SECTIONS. the diagonal of the parallelogram POQE it will bisect the chord PQ^ which is the other diagonal. Hence HO bisecting the chord .PQ and passing through the intersection of the tangents at the extremities of the chord will be parallel to the axis of the curve and therefore to PV: BPVO, RTPO are parallelograms and PT=RO = FV. CoK. 1. If PN be the ordinate from any point P to the axis of the curve, and PT the tangent at that point, we shall have ^r=^iV: Cor. 2, To draw the tangents to the parabola from an external point T, draw TPV parallel to the axis meeting the curve at P; make PF equal to PP, and through Fdraw the chord QVq parallel to the tangent at P: QT, 3' P will be the tangents required. The proposition of this Article' results from the corre- sponding GV. CT= CP^ of the ellipse and hyperbola in the same way that AS— AX results from the relation CS . CX = CA\ 21. QF' = 4 >S^P.PF. We have already seen that if QR be the perpendicular distance of the extremity of a chord Q Vq from its diameter PF QR' = 4^AS.PV. To find the relation between QR and QV draw^ SY the perpendicular on the tangent at P and join AY which will be the tangent at ^. Join. SP. Then the angle SPY=-RPY^ the interior and opposite angle RVQ, since §F is parallel to PF; the right- -^ angled triangles SPY, QVR are similar and QV^ : QR' :: SP' : SY\ But SY' = AS, SP; ON" THE PAftABOLAi -',n ; ; , ^ . . ,- 101 /. SP':SY'::SP':AS. SP::SP:AS; .-. QV':QR'::SF:A8. But QR'=^4^AS.PV, .'. QV^ = 4>SP.PV. 4iSP is called the Parameter of the diameter PV. 22. The length of the focal chord parallel to the tangent atP = 4>S'P. Let SP intersect ^ F in 0. Then Q V being parallel to the tangent at P, makes equal angles with SP and PV; .-. the angle POV=PVO. /. P V= PO, Hence if Qq passed through S we should have PO = SP = PV and QV'=4SP' or QV==2SP, Qq = 48P.U0, But QV' = Q0.q0+V(7, .-. Q0.q0^4SP,U0. 102 • CONIC SECTIONS. If Q' 0(1 be another chord through 0, P' Y its diameter we shall have Q'0.qO = 4^8F' . UO, Hence QO.qO:Q'0.qO::SP:SP\ When Qq, Q'q move parallel to themselves into any- other position, P and P', the extremities of their diameters, remain fixed and >SP, SP' are constant. Hence the ratio QO .qO :Q'0 , q is constdint. Corollary. From the relation QO.qO = 4iSP . UO we see that if a system of parallel chords cut a line drawn from any point in the curve parallel to the axis, the rectangle of their segments made by that line will equal the rectangle of the intercepts on it and a constant line. This property is analogous to that of the hyperbola in Chap. IV. § 11. For by Chap. iv. § 19 we see that when the axis of the hyperbola becomes more and more nearly parallel to a generating line of the cone, the asymptotes become more and more nearly parallel to that line and therefore to the axis. Hence the limiting direction of a line drawn parallel to the asymptote of an hyperbola when it passes into a parabola is the direction parallel to the axis. 24. 8G=SP. If PG be the normal to the parabola at the point P we shall hsiYeSG=:SP. For if PT be the tangent at P, SP = ST; and TPG is a right angle : hence TG is a r it ^~^ g" diameter of the circle with centre S and distance SP or ST: .\SG=:^SP. This corresponds to the proportion that holds, in the EXA3IPLES. 103 other sections of the cone SP :SG::AS: AX: in the parabola AS = AX sind SG==SP, 25. Na = 2AS, Let FN be the ordinate at P; FT, PG the tangent and normal at the same point. Then XG = 2AS. For TN^2TA, TG = 2TS. .', TG-TN=2TS-2TA, or NG = 2AS. Ex^iMPLES. 1. A series of parabolas pass through two given points, and the axis is always parallel to the same line : prove that the focus will lie always on a certain hyperbola. 2. If two confocal parabolas intersect, their common chord passes through the intersection of the directrices, and bisects the angle between them. 3. Two parabolas have a common directrix, prove that their common chord bisects the line joining their foci at right angles. 4. FSQ is a focal chord of a parabola : FA, QA meet directrix in F, Z. Prove that FZ, QY are parallel to the axis. 5. The normals at the extremities of a focal chord intersect in K. KL is perpendicular to the chord, KF parallel to the axis. Prove that F is the middle point of SL. 6. The tangent at F is parallel to the focal chord QSQ' ] PF is its diameter; prove that the normal at F bisects VS. 7. Given two tangents and the directrix, find the focus. 104 EXAMPLES. 8. Tangents at P, Q extremities of a focal chord intersect on tlie directrix in T. Normals at P, Q cut TS in Z7, V, Prove that FQ' = TU. TV. 9. Given the focal chord FSQ and the focus >S, find the vertex. 10. The normals at the extremities of the focal chord PSQ intersect in B. Shew that PP' = SQ . QP. 11. OP, OQ are tangents drawn to the parabola from ; OR parallel to the axis to meet the curve in P. The part of the tangent at P between OP, OQ is bisected at P. 12. OP, OQ are tangents drawn to the parabola from 0, OT perpendicular to OP, cuts the normal at Q in T\ join OS, ST : OST is a right angle. 13. PiV is the ordinate of the point P of the parabola: SY the perpendicular from the focus on the tangent at P : prove YP=YA'. 14.. Find a point such that the tangents from it and the focal distances of the points of contact may be a parallelogram. 15. Two parabolas have the same focus and axes coincident, the line SPQ from, the focus cuts them in Pand Q: prove that the tangents at P and Q are parallel. 16. A circle through S touches the parabola at P. MPK drawn from the directrix parallel to the axis meets the circle again in K : prove that MSK is a right angle. 17. Two parabolas have a common directrix, prove that the common tangents intersect in it. 18. A given straight line is a chord of a parabola and at one end a normal : the axis is given in direction : find the focus and directrix. 19. If the tangent at any point P intersect the latus rectum SL in P, prove SL : SR :: P.V : SP, EXAMPLES. 105 20. FQ is a tangent bounded by tangents OR, OT: PV, QV are drawn parallel to OT, OR: shew that V lies on RT. 21. The locus of the middle points of focal chords is a parabola whose latus rectum = half that of the original curve. 22. PF is the diameter of a focal chord QSQ\ QD meets PV produced at right angles in D: prove that VD= VS, and QS. Q'S=QD\ 23. If the diameter OF of a focal chord meets the directrix in prove that SO^ = 2AS , V. 24. A circle touches the directrix at M and the diameter from M cuts the curve in P : the diameter of the circle = iSP. Shew that the common chord passes through S and that MP pro- duced bisects it. 25. Two parabolas with a common vertex are turned in opposite directions on the same axis, and the focus of one is eight times as distant from the vertex as that of the other : the common tangent is drawn at the vertex. Every tangent to the first para- bola has the part between the tangent at the vertex and the axis bisected by the other. 2^. If a circle touch a parabola at P and cut it in two points Q, R: the tangent at P and the chord QR are equally inclined to the axis : and PO drawn to the point where the axis cuts the circle is parallel to QR, and therefore PQ = OR. 27. In a parabola two chords are equally inclined to the axis; if another parabola passes through the extremities of these chords, it will have its axis at right angles to that of the first parabola. 28. Two parabolas have a common focus and axis ; their vertices are turned in opposite directions : a straight line from S cuts them in P and Q : prove that the tangents at P, Q are at right angles to one another. 29. Given two tangents and their points of contact, determine the focus and vertex. 106 EXAMPLES. 30. OP, OQ are tangents to the parabola from 0, prove OP' : OQ' :: SP : SQ. 31. The lines joining the intersections of the tangents to confocal parabolas drawn to both curves at their points of intersec- tion pass through the common focns. 32. OP, OQ are tangents to the parabola from 0', if the chord PQ meets the directrix in F, prove that OSF is a right angle. If OK be at right angles to the directrix, prove SK is at right angles to PQ. If PQ cuts the axis in N, prove that KN is parallel to OS. If OM be drawn to meet the axis at right angles AA£ = AJSf. 33. OP, OQ are tangents to the parabola from 0. Prove that whenP(2 moves parallel to itself, moves parallel to the axis : when PQ moves round a point in the axis, moves at right angles to the axis: when PQ moves round a point in the directrix, moves in a straight line to S. 34. A line drawn from the focus to meet the tangent at a constant angle, has its point of intersection with it, on one of two fixed tangents. 35. Given one tangent to a parabola, to draw two others which make a given angle with it. In what case is one of the tangents removed to an infinite distance ] 36. BC the portion of a tangent intercepted between two other tangents AB and AC is bisected by D the point of contact. Prove that aS'^ is a fourth proportional to AD, AB and AG. 37. The portion of any tangent between tangents that meet on the directrix, subtends a right angle at the focus. 38. The tangent at P meets the directrix in F : from any point in PF, and from F tangents are drawn to the curve, proVe that they meet in the line through S at right angles to OS. 39. The chord PQ is a normal at P and QR is drawn parallel EXAMPLES. 107 to the axis to meet PP\ tlie double ordinate through P, produced in li : prove that PP' . P'E is constant. 40. The ordinate through the middle point of KG = PG. 41. An ellipse and parabola have a common focus and direc- trix: diagonals of the quadrilateral formed by joining the four points where the tangents at the extremities of the axis major cut the parabola pass through the focus and through the extremities of the axis minor. 42. An hyperbola is confocal with a parabola, and has the tangent at the vertex of the parabola for its nearer directrix. Prove that the tangent to the parabola at tlie point of intersection passes through the further vertex of the hyperbola. Miscellaneous Examples. 1. The orthogonal projection of a parabola is a parabola. 2. The projection of a parabolic section of a cone on a plane at right angles to the axis of the cone is a parabola having for its focus the point where the axis cuts the j)lane on which the projec- tion is made. 3. CS^ the part of the generating line of the cone which has the same projection on the axis as CA has : this was proved for the ellipse, extend the proof to the hyperbola. 4. If an elliptic or hyperbolic section of a cone be projected on a plane through one vertex at right angles to the axis of the cone, CS' is diminished by the square on the distance of C from the plane of projection, and one focus of the projected curve will lie at the intersection of the axis of the cone with this plane. 5. All parabolas cut by parallel planes from a given cone have their foci on a straight line through the vertex of the cone. 108 EXAMPLES. 6. Given a right cone and a point within it ; only two sections have this point for focns and their planes are equally inclined to the line joining the point to the vertex. 7. Given the vertex of a cone and the centre of a sphere inscribed in it : all sections made by planes at right angles to a generating line and to the plane of the paper containing the centre and vertex, will have one of their foci on a circle which i touches the axis of the cone at the centre of the sphere. 8. Two cones touch the same two spheres, prove that by whatever planes the two cones are cut, the ratio of their eccentri- cities is constant. 9. Two cones have supplementary angles and are placed with their vertices and one generating line of each coincident. Curves are cut from them by a plane at right angles to the coincident generating lines : shew that the directrices of either curve pass through the foci of the other. 10. The intersection of a plane with a cylinder is an ellipse with foci at the points of contact of the plane and two spheres in- scribed in the cylinder. CAMBRIDGE: PJillSIED B^ C. J. CLAY, M.A. AX XUE UNIVEUSIXY PRESS. October, 1S84. 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