$B SEfl M7b LIBRARY OF THE University of California. Class A TEEATISE ON GEOMETKICAL CONICS A TEEATISE ON GEOMETEICAL CONICS IN ACCOKDANCE WITH THE SYLLABUS OF THE ASSOCIATION FOIl THE IMPROVEMENT OF GEOMETRICAL TEACHING. BY ARTHUR COCKSHOTT, M.A., ASSISTANT MASTER AT ETON COLLEGE, FORMERLY FELLOW AND ASSISTANT TUTOR OF TRINITY COLLEGE, CAMBRIDGE, AND REV. F. B. WALTERS, M.A., PRINCIPAL OF KING WILLIAM's COLLEGE, ISLE OF MAN, AND FELLOW OF queens' COLLEGE, CAMBRIDGE. llontion: MACMILLAN AND CO. AND NEW YORK. 1889 [All Eights reserved.] CambritigE : FEINTED BY C. J. CLAY, M.A. & SONS, AT THE UNIVERSITY PRESS. PEEFACE. rjlHE need of some lecogDized sequence of" propositions J- in Elementary Geometrical Conies has long been very generally admitted. This need the Association for the Im- provement of Geometrical Teaching has attempted to supply by the publication of the Syllabus of Geometrical Conies, which was drawn up by an influential Committee and ac- cepted by the Association at their annual General Meeting in January, 1884. In the following pages we have given proofs of the propositions in the hope that they may be found useful to those teachers who desire to adopt the order to which the Association has given the weight of its approval. We have introduced a chapter on Orthogonal Projection immediately after that on the Parabola, as we think it important that the student should understand as early as possible the close connection between the ellipse and circle and should be introduced at once to a method by whiclr so VI PREFACE. many properties of the ellipse may be deduced from well- known properties of the circle. At the end of the book will be found a large collection of Cambridge problems ; we have given a list of important properties of Conies, not included in the propositions in the text — all of which are considered as well known and may therefore be assumed in the solution of any other problems. A. C. F. B. W. 3Iay, 1889. TABLE OF CONTENTS. Parabola PAGE . 1 Orthogonal Projections . 26 Ellipse . 33 Hyperbola . 74 Rectangular Hyperbola . 120 Sections of a Cylinder and Cone . . 121 Additional Propositions . 144 Problems . 148 PAEABOLA. Def. I. A parabola is the locus of a point (P), whose distance from a fixed point (>S') is equal to its distance {PM) from a fixed straight line (XM), {SP = Pil/). II. The fixed point {S) is called the focus. III. The fixed straight line (XM) is called the directrix. Def. a curve is symmetrical luitk respect to a straight line, if, corresponding to any point on the curve, there is another point on the curve on the other side of the straight line such that the chord joining them is bisected at right angles by the straight line. Def. The straight line is called an axis of the curve. Def. a vertex is a point at which an axis meets the curve. c. G. PARABOLA. Proposition I. Construction for iwinis on the parabola. The perpendicular on the directrix through the focus is an axis of symmetry. Let S be the focus and MXM' the directrix. Throusfh S draw a straight line SX perpendicular to the directrix, and produce it indefinitely in the direction XS. Bisect SX in A; then because SA =x[X, ^ is a point on the parabola. PARABOLA. 3 In XS or XS produced take any point X ; through X draw a straight line FXP' perpendicular to XX; with centre >S' and radius equal to XX describe a circle, to cut (if possible) PXP' at P and P'; and draw PM, P'M' perpendicular to the directrix. Then because SP = i\^Y = PM, therefore P is a point on the parabola. Similarly P' is a point on the parabola. Since XP = XP\ [Euc. iii. 3. PP' is bisected at right angles by XS, and the curve is symmetrical with respect to X8. (1) If X and''>S» lie on, the same side of A, SX is less than XX, and the circle will cut the line PXF. (2) If X and S lie on opposite sides of A, the circle will not cut the straight line PXP'. • Hence the parabola is unlimited in extent, but lies entirely on one side of a line through A perpendicular to For riders see p. 7. Def. The axis {SX) of a parabola is a straight line through the focus perpendicular to the directrix. Def. The vertex {A) of a parabola is the point at which the axis meets the curve. Def. The ordinate (PX) of a point on a parabola is the perpendicular from the point (P) upon the axis. Def. The abscissa (AX) is the portion of the axis be- tAveen the vertex and the ordinate. Def. The focal distance (SP) of a point on a parabola is its distance from the focus. 1 — 2 PARABOLA Proposition II. If the chord PP' intersects the directrix in K, SK bisects the exterior angle between SP and SP'. Join SP, SP\ Draw PM, P'M' perpendicular to the directrix, and pro- duce PS to p. Then, by similar triangles PKM, FKM', PK : FK = PM : P'M' = SP : SF', :. SK bisects the exterior angle P' Sp. [Euc. VI. A. PARABOLA. Proposition III. //'PN is an ordinate to the jMrabola at the point P, then PN'^ = 4AS.AN. t^ Join SP, and draAv PM perpendicular to the directrix. Then, since >S'^ = ^ A', and AS is divided in iV(Fig. 1), AX is divided in S (Fig. 2) ; .-. NX^ = SX' + 4! AS . AX. [Eiic. ii. S. But A^Y^= P2P = SP' = PX'-\-SX'; .-. PX'+SX' = SX'-h4^AS.AX; :. PX'=4>AS.AX. 6 PARABOLA. Def. The double ordinate through the focus is calle(^ the latus rectum {LL). Proposition IV. Tlie latus rectum LL'= 4AS. SL = 2AS; LL' = 4^AS. [Prop. 3. PARABOLA. 7 PROBLEMS. Prop. I. 1. To trace the parabola by points by means of Euc. i. 2.3. '2. PP\ QQ' are double ordinates to the parabola. Shew that PQ, P'Q' meet the axis in the same point. 3. If SM meets the parallel through A to the directrix in Y, shew that SM is bisected in 1'. 4. Shew also that PY is perpendicular to .S'.V and bisects angle SPM. 5. SZ is drawn i^erpendicular to SP to meet directrix in Z. Shew that PZ bisects the angle SPM. 0. If two focal chords of a parabola are equal, they are equally inclined to the axis. 7. Find locus of centre of a circle which touches a given straight line, and passes through a given point. I/' 8. Find locus of centre of a circle which touches a given circle and a given straight line. 9. A straight line parallel to the axis meets the parabola in one point only. Prop. II. 1. Pj) is a focal chord of a parabola, Q any other point on the curve. If PQ, pQ meet the directrix in K and A" respectively, KSK' is a right angle. 2. PQ, pq are focal chords. Shew that Pp, Qq, meet on the directrix. As also do Pq, pQ. yd. If they meet the directrix in K and A", KSK' is a right angle. 4. Trace the parabola by means of this proposition, by joining A to different points in the directrix. 5. P is any point on the parabola. If PA produced meet the directrix in K, 31 SK is a right angle. 6. Given a parabola and its focus, find the directrix. 7. PQ is a double ordinate of the parabola, PX cuts the curve in P'; prove that P'Q passes through the focus. Prop. III. ' 1. PP' is a double ordinate of the parabola. If the circle round PAP' cut the axis again in Q, shew that NQ is constant and find its length. '~^' 2. PXP' is a double ordinate of the parabola. Through Q, another point on the parabola, straight lines are drawn, one passing tlu'ough the vertex, and the other parallel to the axis, cutting PP' in L and L'. Shew thatxVL..VL' = PX-. Prop. IV. 1. Find a double ordinate PP' of a parabola which shall be double the latus rectum. 2. The radius of the circle described about the triangle LAL' = ^ latus rectum. 8 PARABOLA. Def. Let PP' be the chord of any curve. Then if the point P' move up to P, the chord PP' in the limiting position when P' coincides with P is called the tangent at P. Proposition V. If the tangent at P meets the directrix in Z, PSZ is a right angle, and the tangent at P bisects the angle hetiueen the focal distance SP and the perpendicular PM on the directrix; and the tangent at the vertex is at right angles to the axis. In the figure of ProjD. II. let the chord PP'K become the tangent PZ by moving the point P' up to P, then ultimately SK coincides with SZ, SP' coincides with SP, and the angle P'Sp becomes two right angles ; but P'SK is always half the angle P'Sj) (Prop. 11. ), hence PSZ is half of two right angles, or PSZ is a right angle. PARABOLA. 9 Draw PM perpendicular to the directrix, PM' + MZ' = PZ' = SP' + SZ' ; .-. MZ' = SZ\ since PM = SP ; .-. MZ = SZ; .•. in the triangles ZPM, ZPS, PM, MZ = PS, SZ each to each, and PZ is common to both ; .-. the Single MPZ=SPZ. [Euc. I. 47. [Euc. I. 8. X A If the point P be at the vertex ^4, the angle SPM is two right angles and coincides with the straight angle SAX. Hence the tangent, which bisects tliis angle, is at right angles to the axis. Prove from the definition of a parabola that the straight line which bisects the angle SPM cannot meet the curve in a second point. For riders see p. 11. 10 PARABOLA. Peoposition VI. The tangents at the extremities of a focal chord intersect at right angles on the directrix. Let PSp be a focal chord, and let the tangent at P intersect the directrix in Z. Join Z^S, Zp, and draw PM, pni perpendicular to the directrix. PARABOLA. 11 Then, •. • VZ is the tangent at J?, .'. BZ is at right angles to F^p; [Prop. 5. .•. j)Z is the tangent at p. Again, '.• the A ^FZ= A MPZ, [Euc. i. 4. .-. Z SZP= Z PZM; .-. SZPkhaUofSZM. Similarly SZp is half of SZm, .-. PZp is half of SZM and SZm together, is half of two right angles ; .". PZp is a right angle. PROBLEMS. Prop. V. 1. The tangents at the extremities of the latus rectum meet the directrix at the point A'. 2. If any point be taken on the tangent at P, OM=OS. 3. If the tangents to the parabola at F and P' meet in <), and PM, P'M' be the perpendiculars on the directrix from P and P', OM, OS, OM' are all equal. Deduce a construction for drawing the two tangents from an external point 0. 4. If two tangents OQ, OQ' be drawn to a parabola, and V be the middle point of QQ', prove that OV is parallel to the axis. 5. Hence, given two tangents to a parabola, and their points of con- tact, to find the focus. C), If the tangent at P meet the latus rectum produced in A', and the directrix in Z, SK=SZ. Prop. VI. 1. If the tangents at the extremities of the focal chord PP^ meet in Z and PM, PjiVj be perpendiculars on directrix, shew that JiJ/j is bisected in Z. Hence, prove that the circle described on PP^ as diameter touches directrix in Z. 2. PSQ is a focal chord. QG perpendicular to the tangent at Q cutting axis in G. GZ is a perpendicular on the tangent at P. Shew that Z lies on the latus rectum. 3. Tangents at the extremities of a focal chord cut off equal intercepts on the latus rectum. 12 PARABOLA. Def. The straight line which is drawn through any point on a curve at right angles to the tangent at that point is called the normal. Proposition VII. If the tangent and normal at P meet the axis at T and G respectively, gg ^ gp ^ ST. [Euc. I. 29. [Prop. 6. Draw PM perpendicular to the directrix. Then Z STP = z MPT = z SPT :. SP = ST. And since TPG is a right angle, a circle centre S and distance SP or >ST will pass through G (Euc. ill. 31) ; .". SG = hP = SI . 1. Prove that SM and FT bisect each other at right angles. 2. If T is the middle point of AX, then jV is the middle point of AS. 3. If the triangle SPG is equilateral, the angle TJIG is a right angle. 4. A circle can be described round the quadrilateral SFMZ, and this circle touches FG at P. 5. If the radius of this circle equal MZ, the triangle SFG is equilateral. 6. The angle between any two tangents to a parabola is half the angle which their chord of contact subtends at the focus. 7. The base AB and the angle C of a triangle ABC are given. Find the locas of the focus of a parabola touching CA, CB in .-1 and />. 8. Two parabolas have tlie same focus, and their axes in the same straight line, but in opposite directions. Prove that they intersect at right angles. r ^J PARABOLA. 13 Def. If the tangent and ordinate at the point P meet the axis in T and N respectively, NT is called the subtangent of the point P. j Proposition VIII. Subtcmgent NT = 2 AN. [Prop. 7. Draw PM perpendicular to the directrix. Then ST = SP = PM = XN; and AS = AX, .-. AT = AX; .-. XT = '2 AX. 1. If 7v be the radius of the circle described round the triangle PST. prove that R' = SP . ^^V. 2. From S a line SQ is drawn parallel to the tanpent at P, meeting PE, which is parallel to the axis, in E. Shew that the locus of £ is a parabola, vertex S and latus rectum = h that of original jiarabola. 14 PARABOLA. Def, If the normal and ordinate at the point P meet the axis in the points G and N respectively, NG is called the subnormal of P. Subnormal Proposition IX. NG = 2AS. [Prop. 7 Draw PM perpendicular to the directrix. Then SG = SP = P3I = XN; .-. NCr = SX = 2 AS. 1. If the triangle .SPG is equilateral, 5P = latus rectum. 2. Deduce Proposition -4 from Propositions 8 and 9. 3. To draw the normal to the curve at any given point. 4. If QM, the ordinate of Q, bisect NG, prove that QM=PG. 5. TP, TQ are tangents to a given circle. Construct a parabola which shall touch TF in P and have TQ for axis. PARABOLA. 15 Proposition X. If the tangent at an 1/ point P intersects the tangent at the vertex iii Y, then SY bisects PT at right angles, and is a mean pivportional between SA and SP (SY'^ = AS .SP). Join SP, and draw PN perpendicular to the axis. Then, since T^Y is bisected in A, and A Fis parallel to PN, .'. PT is bisected in Y. The angles SYT, SYP are equal; [Euc. i. 8. .•. >SF is at rio[ht angles to PT. Again, because YA is drawn from the right angle per- pendicular to the base ST of the triangle SYT, .-. SY' = SA.ST [Euc. VI. 8. = SA.SP. [Prop. 7. 1. The circle on SP as diameter touches the tangent at the vertex in Y. 2. Vro\e FY. PZ = SP'. 3. Vro\ePY.YZ = AS.SP. 4. SY produced meets the directrix in M. 5. If a circle be described on the latus rectum as diameter, and PQ be a common tangent to the parabola and circle, touching them in P and Q respectively, shew that SP, SQ are each inclined to the latus rectum at an angle of 30^. 6. Given two tangents to a parabola and the focus, shew how to draw the tangent at the vertex, and hence the axis and directrix of the parabola. 7. A long rectangular slip of paper is folded so that one of the corners always lies on the opposite side. Prove that the crease always touches a parabola, of which the opposite side is the directrix. 16 PARABOLA. Proposition XI. If from any point on the tangent at P, 01 is drawn 2)erpendicular to the directrix, and OU perpendicular to SP, then SU = 01. (Adams's property.) I ^..^-^ ^^^X^^ / Z ^s/ Xj T \ ^ Join HZ, and draw FM perpendicular to the directrix. Then, since angle ZHF is a right angle, .'. ZB is parallel to OU. .-. m : 8P = Z0 : ZP = 01 : PM. But SP = P3I; .-. SU=OL PARABOLA. Proposition XII. 17 To draw tiuu tanrjents to the parabola front an external point O. (Analysis. Let OQ, OQ' be the two taugents. Draw QM, Q'M' per- pendiculars on the directrix, and join OS, 0^1, OM'. Then, since the angle SQM is bisected' by OQ, therefore the triangles SQO, MQO are equal (Euc. I. 4) and 0M= OS. So Oil/' = OS. Thus the points M and M' are found, hence construction.) With centre at distance OS describe a circle, cuttin*:- the directrix in M and iV. From M and M' draw MQ, M'Q' to the parabola, at right angles to the directrix. Join OQ, OQ' . OQ, OQ' shall be the tangents required. Join OS, OM, OM', SQ, SQ'. Then, in the triangles SQO, MQO, SQ, QO = MQ, QO, and the base 0.1/= base OS] .-. the angle SQO = angle MQO; .'. OQ is the tangent at Q. [Prop. 5. So OQ' is the tangent at Q'. Note. The construction may be made on the principles proved in Propositions 10 or 11. For riders see p. 25. C. G. 2 18 PARABOLA. Proposition XIII. The two tangents OQ, OQ' subtend equal angles at the focus, and the triangles SOQ, SQ'O are similar. Draw the tangent at the vertex, meeting OQ, OQ' in Y and Y. Join SQ, SQ', SY, SY. Produce QO to meet the axis in T. Then, since the angles at Y and Y' are right angles, [Prop. 10. the circle on OS as diameter will pass through Y and Y\ Therefore angle SOQ' = angle 8YY' in same segment = angle STY [Euc. vi. 8. = angle SQO. [Prop. 7 and Euc. I. 5. Similarly angle SQ'O = angle SOQ ; .* . remaining angles OSQ, OSQ' are equal, and the triangles SOQ, SQ'O are similar. OS and a line through parallel to axis make equal angles with the tangents. For riders see p. 25. PARABOLA. Proposition XIV. 19 If a pair oj tanrjents OQ, OQ' cu^e drawn to a parabola, and OV is drawn parallel to the axis, meeting QQ' in V, QQ' luill he hii^ected in V. o M ^^^ f-f^^ ^ / / / /v \\ ^^~^\^ ^ • M' Q\. Let V cut the directrix in R. Draw QM, Q'M' perpendicular to the directrix. Join Oil/, OS, OM', SQ, SQ'. Then, in the triangles SQO, MQO, SQ, QO = MQ, QO, and angle SQO = angle MQO ; [Prop. 5. .-. OM=OS. Similarly OM' = OS ; .-. OM=OM', and OR, which is drawn at right angles to the base of the isosceles triangle OMM', bisects it; .-. MR = M'R. But QV : Q'V=MR : M'R; .-. QV= Q'V, or QQ' is bisected in V. For riders see p. 25. 2—2 20 PAEABOLA. Proposition XV. The locus of the middle points of any system of parallel chords of a parabola is a straight line parallel to the axis. And the tangent at its point of intersection ivith the parabola is parallel to the chords. Let QQ' be one of the chords, and RPR' the tangent parallel to them, touching the parabola at a fixed point P. Through P draw OPV parallel to the axis, meeting QQ' at V and the tangent QRO at 0. Join PQ and draw RW parallel to the axis, bisecting PQ at W. [Prop. 14. Then OR = RQ because RW is parallel to OP, [Euc. VI. 2. and OP = PV because PR is parallel toQV. Similarly if we draw a tangent Q'R'O' meeting OPV at 0', OP = PV, hence and 0' are coincident. Since OQ, OQ' are tangents and OF is parallel to axis, QQ' is bisected at V. [Prop. 14. Hence the locus of the middle points of all chords parallel to RPR! is a straight line through P parallel to the axis. Def. The locus of the middle points of any system of parallel chords drawn in a curve is called a diameter. NoTK. A diameter will not be a straight line for all curves. It has just been proved to be so for a parabola. For riders see p. 25. PARABOLA. 21 Def. The half ch(jrds (Q V) intercepted between the dia- meter and the curve are called ordinutes to the diameter. Proposition XVI. If QV ^^ ^^'^ ordinate of a diameter PV, and the tangent at Q meets VP produced in O, then OP = PV. Draw PR touching the parabola at P and meeting OQ at R ; through R draw R W parallel to the axis. Since RP, RQ are a pair of tangents, PQ is bisected at TF, and PR is parallel to QV; ... OP : PV=OR : RQ = PW : WQ. But PW=WQ, .'. OP = PV. [Prop. 14. [Prop. 15. 22 PARABOLA. Proposition XVII. //QV is an ordinate to the diameter PV, then Qy2 = 4sp.pv. Let the diameter PV meet the parabola in P. Draw the tangent at Q, meeting the diameter in and the axis in T. Draw the tangent at P, meeting OQ in R. Join SP, 8R, SQ. Then, since RP, RQ are two tangents, .-. the triangles SRP, SQR are similar ; .-. the angle SRP = angle SQR = angle STR = angle POR, and the angle SPR = angle OPR. '.' the tangent at P bisects the angle SPO, [Projj. 5. .'. the triangles SRP, POR are similar. .-. PR' = SP .PO. . Now OV is bisected in P (Prop. IG), .'. QV=2PR, ,', QV'=4'PR^ = 4. [Euc. III. 31. [Def. of parabola. [Prop. IG. 24 PARABOLA. Proposition XIX. If ttuo chords, QQ', qq', of a parabola intersect one another, the rectangles contained hy their segments are in the ratio of the parallel focal chords ; or QO . Q'O : qO . q = 4SP : 4Sp. Draw the diameter P V to bisect QQ' in V. Draw OTT parallel to the axis, to meet the parabola in W. Draw the ordinate WR to the diameter PV. Join SP. Then Similarly QO . Q = QV - OV ^QV^-WR' = 4'SP.PV-4'SP.PR = 4^SP .RV = 4>SfP . W. qO .qV = 4'Sp, OW; Q'O : qO . q'O = 4.SfP : hSp. For riders see p. 2G. [Euc. II. 5. [Euc. I. 34. [Prop. 16. QO PARABOLA. 25 Prop. XTT. 1. If the point be on the directrix, shew from the construction that the tangents intersect at right angles. 2. Find the point O so that the figure OQSQ' may be a parallelogi-am. Prop. XIII. 1. If a third tangent be drawn cutting OQ, OQ' in B. and T, prove that the circle which circumscribes the triangle ORT will pass through H. 2. "What is the locus of the focus of a parabola which touches three given straight lines ? 3. A parabola touches each of four straight lines given in position. Give a Geometrical construction for finding its focus. •i. Prove that O.S is a mean proportional between 0(^ and OQ'. WTiat previous proposition is a particular case of this? 5. Two tangents to a parabola and the point of contact of one of them are given. Shew that the locus of the focus is a circle passing through the given i^oint of contact and the intersection of the tangents, and touching one of them. G. The straight line which bisects the angle QOQ' between the two tangents meets tlie axis in 11. Shew that SO = Sll. Prop. XIV. 1. The circle on any focal chord as diameter touches the directrix. 2. The normals at the extremities of a focal chord intersect on the diameter which bisects the chord. 3. Given two tangents and their points of contact, to find the focus and directrix. Prop. XV. 1. Tangents at the extremities of all parallel chords meet on the same straight line. 2. A parabola being traced on paper, find its axis and directrix. 3. If a chord make an angle of 45° with the axis, the line through their middle points passes through an extremity of the latus rectum. Prop. XVU. 1. If QD be drawn perpendicular to OV, Qiy- = iAS . PV. 2. If TPV is diameter at P, QV an ordinate, and QT tangent at Q, and if QV= TV, shew that T is on the directrix. 3. Any chord LVL' is drawn through V, and LM, L'M' are the ordinates of LL' drawn to the diameter P I '. Prove that L^f . L'M = Q\"-. 26 PAEABOLA. 4. If from the point of contact of a tangent to the parabola a chord be drawn, and another line be drawn parallel to the axis, meeting the tangent, curve, and chord, this line will be divided by them in the same ratio as it divides the chord. 5. Draw a chord of a parabola through a given point, so as to be cut in a given ratio at the point. Prop. XVIII. 1. To draw a focal chord PSQ such that SP=SSQ. 2. If a diameter meet the directrix in 0, OS is perpendicular to the chords bisected by the diameter. Prop. XIX. 1. The semi latus rectum is a harmonic mean between the segments of any focal chord. 2. If QVhe an ordinate to the diameter PV, and_2w meeting PQ in v be the diameter conjugate to PQ, then jJv - \P V. ORTHOGONAL PROJECTIONS. Def. I. If from any point a perpendicular be drawn to a fixed plane, the foot of the perpendicular is called the projection of the jmint, and the fixed plane is called the plane of projection. II. The projection of a line, straight or curved, is the aggregate of the projections of its points, that is the locus of the feet of perpendiculars, drawn from points on the line, to the plane of projection. III. Tlie projection of an area is the area contained by the projection of the line or lines containing the given area. IV. The straight line, in which the plane, containing a given curve, intersects the plane of projection, is called the base line. ORTHOGONAL PROJECTIONS. 27 Proposition i. The 'projection of a straight line is a straight line. Let pqrsU be the given straight line meeting the base line in U, and let P, Q, R, S be the projections oi p, q, r, s. Then the perpendiculars pP, qQ, rR, sS will lie in one plane pPU (Euc. XI. 6, 7) which intersects the plane of pro- jection in a straight line UP (Euc. xi. 3). Hence the projection of Up is the straight line UP, and they intersect in a point U on the base line. Proposition ^. The ratio of the segments of a finite straight line is unaltered by projection. Let pqrsU be the given straight line, and PQRSU its projection. Then pP, qQ, rR, sS are parallel because they are all perpendicular to the plane of projection, and they are all in the same plane P Up ; hence the segments PQ, QR, RS are in the same ratio as pq, qr, rs (Euc. vi. 2). 28 ORTHOGONAL PROJECTIONS. Proposition 7. Parallel straight lines project into j'x^vallel straight lines of jyroj^ortional length. Let pqU, rsV be two parallel straight lines, meeting the base line in ?7and V, and let PQU, RSV he their projections. 2)P and rR are parallel, [Euc. xi. 6. 2Jq and rs are parallel ; [l^yP- .'. the plane f/pP is parallel to plane VrR. [Euc. XI. 15. Hence PQUis parallel to RSV. [Euc. xi. 16. Again, triangles pUP, rVR are equiangular, [Euc. xi. 10. .-. PQ : pq = PU : pU, = RV : rV, = RS : rs. Obs.— This ratio PU :pU = cos pUP. ORTHOGONAL PROJECTIONS. 29 Proposition 8. A tangent projects into a tangent, cutting the base line in the same point. Let 2^P ^G two points on a curve near to one another, then their projections PF' lie on the projection of the given curve. Let p move up to and coincide with p, so that pp becomes a tangent to the given curve. Then P' moves up to and coincides with P, and PP' becomes a tangent to the projection of the given curve. Also these straight lines meet the base line in the same point. (Prop, a) 30 ORTHOGONAL PROJECTIONS. Proposition e. The ratio of areas is unaltered hy 'projection. Case 1. Let pqrs be a rectangle, having two sides pq, rs parallel to the base line, and let PQRS be its projection ; produce ps, qr to meet the base line in U, V. Area PQRS : area pqrs = PQ x PS : pq x ps, = PS : ps, = PU : pU. Now this ratio (which is equal to cos a, if a be the angle between the original plane and the plane of projection) is independent of the length and breadth of the rectangle ; therefore all such rectangles are diminished by projection in the same proportion, and all such rectangles drawn in the original plane bear the same ratio to one another as their projections do. ORTHOGONAL PROJECTIONS. 31 Case 2. But a figure of any shape may be divided into a large number of narrow strips by lines perpendicular to the base line, and each of these strips will form one of these rectangles, with two small areas at each end ; now the sum of these rectangles bears to the sum of their projections a constant ratio, also by increasing the number of rectangles and decreasing their width the difference between them and the given area may be indefinitely diminished, hence an area of any shape is diminished by projection in the same ratio (1 : cos a) and all areas in the original plane bear the same ratio to one another as their projections do. 32 ORTHOGONAL PROJECTIONS. Proposition f The projections of two straight lines at right angles to one another are lines at right angles to one another, if one of the original lines is parallel to the base line. Let ps, sr be two straight lines at right angles to one another, of which sr is parallel to the base line UV. Let PS, SR be their projections. Since sr is parallel to UV, it does not meet the plane of projection PSUV, hence sr does not meet SR ; also sr, SR are in the same plane, therefore they are parallel to one another. But SR is at right angles to Ss, therefore sr is at right angles to Ss', also sr is at right angles to ps, :. sr is at right angles to the plane psUSP ; SR is at right angles to the plane ps USP, and PSR is a right angle. [Euc. I. 21). [hyp- [Euc. XI. 4. [Euc. XI. N. Note. The projection of a right angle is not a right angle, unless one of the arms of the original angle is parallel to the base line. ELLIPSE. Def. I. An ellipse is the locus of a poiut (P) whose distance from a fixed point (S) bears a constant ratio (e), less than unity, to its distance (PM) from a fixed straight line (Zil/), {SP = e . PM). II. The fixed point (S) is called the focus. III. The fixed straight line (XM) is called the directmx. IV. The constant ratio (e) is called the eccentricity. . C. G. 34 ELLIPSE. Proposition I. Construction for jjoints on the ellipse. The i^erpendicular on the directrix through the focus is an aocis of symmetrij. To find the vertices A and A'. From the focus 8 draw SX iDerj^endicular to the directrix. Divide XS in J., so that SA = e.AX', also in XS produced take A' so that SA' =e.A'X. Then A and A' are points on the curve. Take any point N on the straight line AA\ with centre 8 and radius e. XN describe a circle; through N draw PNP' perpendicular to AA' and cutting the circle in P and P', then P and F are points on the ellipse. Draw PAI^ P'M' perpendicular to the directrix, 8P = e.XN = e.PM, 8P'=e.XN=e.P'M'. Corresponding to any point N on the line AA\ we thus get two points P and P' at equal distances on opposite sides of AA' ) hence the ellipse is symmetrical with respect to AA', or AA' is an axis, and the points A and J.'*are vertices. Note, It may be proved that the circle intersects the perpendicular NP, when N is any ])art of tlie axis A A' between .1 and A\ but not when N lies outside the part AA\ hence the elliiise lies entirely between lines drawn through A and A' at right angles to the axis. See Appendix. For riders see p. 87. ELLIPSE. 35 Proposition II. If the chord PP' intersects the directrix in K, SK bisects the exterior angle hetiveen SP and SP'. Join Sl\ SP', SK\ produce PS to ;;, and draw PM, P'M' perpendicular to the directrix. Then SP = e. PM, and SP' = e . P'M' ; .-. SP : SP' = PM : P'M' = PK : P'K, by similar triangles PKM, P' KM'. Therefore SK bisects FSp (Euc. v!. A.). Prop. II. 1. P.SPi is a focal chord. Prove that XP and A'/'j are eiiually inclined to the axis. 2. PNP^ is a focal chord. PA, P^A are produced to meet the directrix in A' and A'^ respectively. Prove that A'.SA'j is a right angle. 3. Two chords Pp, P'Q meet the directrix iu p, p' respectively. Prove that the angle pSp' is half the angle PSV. 4. If the focus of an ellipsfe and two points on the curve be given, the directrix will pass through a tixed point. '^—■1 36 ELLIPSE. Def. If the axis throiigli the focus (S) meets the ellipse at A and A\ AA' is called the major axis. Def. Bisect AA' in C, then C is called the centre of the ellipse. Def. The double ordinate BCB' , drawn through C, is called the minor axis. Proposition III. If PN is the ordinate of a point P on the ellipse, VW : AN . A'N = CB^ : CA^ and CB is less than CA. Join PA, A'P, and produce them to meet the directrix at K and K'. Join SP, SK, SK'y and produce PS to 2)- By similar triangles PAN, KAX, PN : AN = KX : AX. By similar triangles PA'N, K'A'X, PN : A'N=^1CX : A'X; .-. PN': AN.A'N=KX.K'X : AX.A'X. ELLIPSE. o7 But SK bisects the angle ASp, [Prop. 2. and 8K' bisects the angle ASP, [Prop. 2. .'. KSK' is a right angle ; .-. KX . K'X = SX' ; [Euc. vi. n. .-. PN': AN.A'X=SX': AX.A'X. Similarly, since P may coincide with B, BC : AC = SX' : AX .A'X, .-. PX' : AN.A'X= BC : ACr\ Again, BC : AC = SX' : AX . A'X. Now SX = AX-\- SA = AX (1 + e), also SX = A'X - SA' =A'X (1 - e), .-. SX' = (1 - e'} AX . ^'X < ^.Y . A'X ; .-. BCS^' as focus, and eccentricity the same as before. Prop. IV. 1. A straight line cannot meet the elHpse in more than two points. 2. Of all lines drawn from the centre to the curve CA is the greatest and CB the least. 3. P and Q arc corresponding points on the ellipse and the auxiliary circle; through P KVL is drawn making the same angle with the axes which CQ does, and cutting them in K and L. Shew that KL is a constant length. 4. PM drawn perpendicular to BW meets the circle on the minor axis as diameter in p'. Prove PM :p'M=CA : CB. 5. If the two extremities of a rod slide along two fixed straight lines at right angles to one another, any fixed point in the rod will describe an ellipse. Prop. V. An ellipse may also be itself projected into a circle. 42 ELLIPSE. Proposition VI. (Aliter.) Let aha be a circle, and ABA' its projection. All chords of the circle parallel to aa are bisected by cb. [Euc. IIL 3. Therefore all chords of the ellipse parallel to AA' are bisected by CB. [Prop. 7. And CB is perpendicular to chords it bisects. [Prop. ^. Hence the ellipse is symmetrica] with respect to the minor axis. And it may be described with reference to a second focus and directrix on the opposite side of the centre. ELLIPSE. 43 Proposition VII. CA = e . CX ; CS = e . CA ; CS . CX = CA^ SA = e.AX, [Dei'. SA' = e.A'X. [Def. By addition AA' = e {AX + A'X) = e {AX + AX) = eXX'; .: CA = e.CX (2). By subtraction SS' = e.AA'; .-. CS = e.CA (/9); .-. CS.CX=CA' (7). Prop. VII. Given an ellipse and one focus, find the centre and the eccentricity. 44 ELLIPSE. Proposition YIII. SP + S'P = AA'. Mechanical construction for the ellipse. Draw MPM' perpendicular to the directrices. Then SP = e . PM, and 8'P = e.PM'; .: SP-[-S'P = e. MM' = e.XX' = AA\ If an endless string be placed round two drawing-pins at S and S', and kept tight by a pencil point at P, the pencil can be made to trace out an ellipse of which >S^, S' are the foci. Pitop. VIII. 1. If P be any j>oint, SP + S'P is greater than, equal to, or less than AA\ according as P is without, upon, or within the ellipse. 2. A circle is drawn entirely within another circle. Prove that the locus of a point equidistant from the circumferences of these two circles is an ellipse. 3. Two ellipses have a common focus, and their major axes equal. Prove that they cannot intersect in more than two points. 4. Prove that the straight line, which bisects the exterior angle between PS and PS', cannot meet the ellipse again. ELLIPSE. 4.5 Proposition IX. CB^ = CA^ - CS' = S A . SA'. But SB + S'B = AA'. Prop. S SB = SB ; Eiic. I. 4 .-. SB=CA, GB' = SB' -GS' "Euc. I. 47 = CA' -GS" = SA. SA\ [Euc. II. 5 46 ELLIPSE. Def. The double ordinate through the focus is called the latus rectum (LL). Proposition X. The semi latus rectum SL is a third iwoportional to CA and CB. .SL.GA = CB\ But SU : AS.A'S=CB' : CA\ AS.A'S Sr : CB' CB'; CB' : CA'; SL : CB=CB : CA ; . SL . CA = CB\ [Prop. 3. [Prop. 0. ELLIPSE. 47 Proposition XI. If the tangent at P meets the directrix in Z, PSZ is a right angle. Also tangents at the ends of a focal chord intersect on the directrix. Take a point P' on the ellipse near to P, and let the chord PP' meet the directrix in K, and produce PS to p. Then KS bisects the angle P'Sp. [Pr(»p. '2. When P' coincides with P, so that PP' K becomes the tangent PZ, P'Sp becomes two right angles; therefore PSZ is a right angle. Hence ZSp is a right angle, and Zp is the tangent at ;>. or the tangents at P and /) intersect on the directrix. 1. Tangents at the extremities of the latus rectum intersect in X. 2. If throiagli any point P of an ellipse (^I*X be drawn perpendicular to the axis, moetmg the tangent at L in (^ and axis in .V, QX = Sr. 3. To draw the tangent at a given point P of the ellipse. 4. By drawing the tangent at B, prove CS . CX=CA'^. 48 ELLIPSE. Proposition XII. If the normal at P intersects the major axis in G, SG = e.SP. Draw the tangent PZ, join 8Z, draw PM perpendicular to the directrix, and join SM. ZMP and ZSP are right angles ; [Prop. 11. therefore the circle, on ZP as diameter, passes through M and >S'. [Euc. ill. 31. Since ZPG is a right angle, PG touches the circle. [Euc. III. 16. Therefore the angle >ST(7 = angle SMP in the alternate segment. [Euc. ill. 32. Also angle PSG = angle SPM. [Euc. I. 29. Therefore the triangles SPG, SMP are similar; ... SG : SP = SP : PiM; .-. SG = e. SP. Prop. XII. 1. P is any point on the ellipse, M a fixed point on the major axis. A perpendicular is drawn from il/ on the tangent at P. Find the locus of the intersection of this perpendicular with the radius vector SP. 2. If GL be drawn perpendicular to SP, the ratio PN : GL is constant, and PL = semi latus rectum. 3. If PG be produced to meet the minor axis in (f, gS produced meets the directrix in M, the foot of the perpendicular from P. ELLIPSE. 49 Proposition XIII. The tangent and normal to an ellipse at any jyoint P are respectiveli/ the external and internal bisectors of the angle between tit e focal distances. Let TPy be the tangent and PG the normal, SG = e.SP, [Prop. 12. and S G = e . S P ; .-. SG : S'G = SP : S'P; therefore PG bisects the angle SPS'. [Euc. vi. 3. Therefore the coniijlements SPT, SPY' are equal, but S'Pr= WPT; [Euc. L 15. therefore PI' bisects the exterior angle SPW. Prop. XIII. 1. If .ST, the perpendicular ou the tangent at P, meet S'P produced in x, prove (1) sY=SY, (2) SP = Ps, (3) S's = AA'. If P move round the eUipse what is the locus of s? [Note. On account of (1) .s is called the image of the focus in the tangent.] 2. If the tangent and normal meet the minor axis in t and g respectively, the circle ou at as diaui tor i)assos through P and the two foci. 3. If the normal at P meet the major and minor axes in G and g, prove that the triangles SPG, pPS' are similar. 4. SP . S'P = PG . Pfj. 5. No normal can pass through the centre, except the normals at the ends of tlie axes. 6. If a circle be described through the foci of an ellipse, a straight line drawn from its intersection with the minor axis to its intersection with the ellipse will touch the ellipse. C. G. 4 50 ELLIPSE. Proposition XIV. Tlte feet ofthepeiyendiculars (SY, S'Y') fi'oDi the foci on the tangent at P ai^e on the auxiliary circle. Also if CE, jmraUel to the tangent at P, intersects S'P in E, PE = C A. Also SY . ST' = CB\ Produce S'P, SY to meet in W. Join CY. In the triangles YPS, YPW, YP is common, rioht ani;les PF>Sf,PyF are equal, angle FPASf = angle YPW; ^[Prop.'^lS. .-. SP =PW, SY=YW; [Euc. l 26. and SG=^CS\ .'. S'W is parallel to CY; .'. CY=hS'W = ^(S'P-\-PS)=iAA' = CA ; therefore Y is on the auxiliary circle. Similarly, Y' is on the auxiliary circle. Also YCEP is a parallelogram ; therefore PE=CY=CA. Produce Y'S' to meet the circle in y and join Yy. Then, YY'y being a right angle, Yy passes through the [Euc. VL 2. [Euc. VL 4. [Prop. 8. centre C, SY=S'y, S Y . S' Y' = Sy . S' F'= AS'.S'A' = CB\ For riders see page 52. [Euc. IIL 81. [Euc. I. 4. [Euc. III. 8;"). [Prop. I). ELLIRSE. 51 Proposition XV. Corresponding chords of the ellipse and aiuilianj circle intersect on the major axis. Also tdnrjents at corresponding points intersect on tJie major axis. q XT FN PX BC\ [Euc. VI. 4. [Eiic. VI. 4. [Prop. 4. Let PQ be a chord of an ellipse, meeting the major axis in T. Let p be the point of the auxiliary circle correspoiulinn to P. Join Tp, and produce it to meet the ordinate R(^) produced in q. Then qR : pX=RT = QR .'. qR : QR=pX = AC .'. q is the corresponding' point to Q, and the correspond- ing cliords PQ, pq meet the axis in the same point T. If Q moves up to and coincides with P, then q niuves up to and coincides with y>, and Pl\ pT become tangents to the ellipse and circle, or the tangents at corresponding points intersect on the major axis. Prop. XV. 1. Pp are corresponding points. The tangent at p meets CB produced in A'. Prove C7v . FN = AC . BC. *2. OQ, OQ' are tangents to an ellipse. O.V is drawn perpendicular to the axis. Prove that the tangents to the auxiliary circle at the corresi^onding points q and q' meet in OX. Prove also that if QQ' produced meet the major axis in T, CX.CT=CA-. 4—2 52 ELLIPSE. Proposition XVI. If the tangent at P meets the 7najor axis produced at T, CN . CT = CA'^ Produce NP to meet the auxiliary circle in j^, and join pT, pG. pT touches the circle; [Prop. XV. therefore CpT is a right angle; [Euc. ill. 18. .-. CN.GT=Cf [Euc. vl8. = CA\ Pkop. XIV. 1. To draw a tangent to the ellipse parallel to a given straight line. 2. If a straight line through C parallel to the tangent intersect the SF, S'P distances in E, E', prove GE = CE'. 3. Prove also S:'E = Sf£'. 4. The circle described on SP as diameter touches the auxiliary circle. 5. HK is parallel to S'P, and YK perpendicular to SK. Shew that the parabola having S for focus and K for vertex touches the ellipse. 6. Given in position a focus and tangent, and in magnitude the minor axis, find the locus of the other focus. 7. A chord of a circle which subtends a right angle at a fixed point envelopes a conic whose foci are the fixed point and the centre of the circle. 8. If a second tangent intersect YPY' at right angles in O, prove that or. OY' = BC\ Hence prove CO'^=CA'^+ GB'-. [The locus of the intersection of tangents at right angles is called the Director Circle.] ELLIPSE. Proposition' XV I. (Aliter.) Draw tlie circle from which the ellipse is projected, nnd let (7, P, T, N, A be the projections of c, 2J, t, ??, (I. Then j)f touches the circle ; therefore cpt is a right angle, and cnp is a right angle ; .*. en . ct = cj/ ; .'. c?i . ct = cir ; .-. CN.CT=CA\ [Prop. ^. [Euc. III. IS. [Prnp. f. [Euc. VI. S. [Pro]). (3. Prop. XVI. 1. p is the point on the auxiliary circle correspomling to P. Sy is drawn periioudicular to the tangent at jf^. Prove Sy = SP. 2. Any circle through N, T, cuts the auxiliary circle at right angles. 3. If CY, AZ he the perpendiculars from the centre and an extremity of the major axis on the tangent to the ellipse at any point P, shew that CA . AZ=CY . AN. o4 ELLIPSE. Proposition XVII. If the tangent at P meets the minor axis produced in t, and Pn is the jyerpendicidar from P on the minor axis Cn . Ct = CBl Draw the circle of which the ellipse is the projection. And let c, p, t' , h, n be the points of which C, P, t, B, n are the projections. Join cp. Then pt' touches the circle ; therefore cpf is a right angle. Also cnp is a right angle ; . *. on' . ct' = cp^ = cIj'; .-. Cv . Ct = CB\ [Prop. S. [Euc. III. 18. [Prop f. [Euc. VL 8. [Prop. ^. ELLIPSE. oo Proposition XVIII. If PF is the perpendicular^ from P on a line through (J parallel to the tamient at P, and if the normal at P meets the minor axis in £j, then PF.PG^CB'^ and rF.Pg = ('A^ Draw PXIi, Pin- perpendicular to the axes meeting CF in B anil r, and let the tangent at F meet the axes at T and t Since the angles at X and F are right angles, a circle can be described tlirough GXR and F; .: FF.PG = PX.PR = On . Ct Similarly = cir-. PF.Fij = Fn.Fr = ex. CT = CA\ [Euc. III. :;i [Euc. III. .S(i. [Euc. I. :U. [Prop. XVII. [Euc. I. :U. Prop. XVIII. 1. If from // a perpendicular gK be dropped on SP or ST, prove tliat PK=CA. 2. If the tan?ent at P meets the major axis in T, then CF . PT is equal to the product of peri>ejidiculars from the foci on the normal at P. Also Proposition XIX. GN : CN = CB'"^ : CA'^ CG = e^CN. B Produce PG to meet the minor axis at cf, and draw CF parallel to the tangent at P, meeting Pc/ at F. Then GN : CN=PG : Pg [Euc. vi. 2. = PF.PG : I'F.Pg = CB' : CA\ [Prop, xviii. Also CX - GN : CN = GA' - CB' : CA' ; .-. CG : CN = CS' : CA'; [Prop, i x . .-. CG = e\CN. [Prop. VII. Puop. XIX. 1. If the tanp^ent and normal at P meet the major and minor axes resix'ctively in 2\ t, G, p, prove (a) CG . CT=CS', (6) Gg.Ct=CS'\ (c) Tg, tG are at right angles. 2. Prove NG . CT=CB^. a. From this proposition deduce the corresponding proposition for the parabola, viz. NG = 2AS. 4. Find a point P on the ellipse such that PG bisects the angle between CP and P.V. ELLIPSE. n( Proposition XX. If from anij point O on the tancjetd at l\ Ol Is drawn perpendicular to the directrix, and OU perpendictdar to SP, then 8U = e . 01. (Adams's property.) Join SZ, and draw PM perpendicular to the directrix. ZSP is a right angle ; [Prop. xr. .'. ZS is parallel to OC ; .'. SU : SP = ZO : ZP [Euc. vi. 2. = 01 : PM, [Euc. VI. 4. but SP = e.PM; .-. SU=e.OI. If the tangent at P meet the directrices in Z, Z', the perpendiculars from Z and Z' on SP intercept a part equal to A A'. 58 ELLIPSE. Proposition XXI. To draiv a pair of tangents OQ, OQ' to an ellipse from an external point O. Draw 01 perpendicular to the directrix. With centre S, and radius e . 01 describe a circle, and draw the tangents OU, U' . [Euc. in. 17. Draw SZ perpendicular to 8U^ meeting the directrix in Z. Join ZO^ meeting >S'f7 in Q. Draw QN perpendicular to the directrix. [Euc. vl 2. Then SQ:SU=QZ:OZ = QN : 01 ; .-. SQ:QN:=SU:OI = e: 1; therefore S is on the ellipse. And since QSZ is a right angle, OQ touches the ellipse. [Prop. 11. Similarly a second tangent OQ' may be drawn. ELLIPSE. 59 {Second method.) On OS as diameter describe a circle meeting the auxil(^y circle in Y and Y'. Then SYO is a right angle [Euc. ill. :U], and OF touches the ellipse [Pm]). XIV.]. Siniil.irly OY' touches the ellipse. 11/ (Third method.) With centre and radius OaS' describe a circle, and with centre S' and radius ^4^' describe a second circle intersecting the first in U and U\ Join S'U, S' U' meeting the ellipse in Q and Q\ then angle (V(^r= angle OQN, [Euc. i. s. and OQ touches the ellipse. [Prop. Xiii. Similarly OQ' touches the ellipse. 60 ELLIPSE. Proposition XXII. Tangents OQ, OQ' subtend equal angles OSQ, OSQ' at the focus S. Draw OU, OU', 01 perpendicular to SQ, SQ', and the directrix. Join OS. Then or SU = e.OI = SU'; OU=OU'; OSU=OSU', OSQ = OSQ'. [Prop. XX. [Prop. XX. [Eiic. I. 47. [Euc. I. 8. Prop. XXII. 1. QQ' produced meets the directrix in A', prove that OSK is a right angle. 2. Tangents at the extremities of a focal chord meet the tangent at the vertex in '1\, 1\,, prove Al\ . A1\^AS-. 3. OQ, OQ' are two fixed tangents to an elHpse. A variable tangent intersects them in rj, q'. Prove that the angle qSq' is constant. 4. Normals at the extremities of a focal chord meet in ir, and the cor- responding tangents in Z. Prove that ZW passes through the other focus. 5. OQ, OQ' are tangents from O, and OS meets QQ' in 7?, ]iZ, parallel to the axis, meets the directrix in Z. Shew tluit QZ an. I Q'Z are equally inclined to the axis. ELLIPSE. (U Proposition XXllI. OS'. Tangents OQ, OQ' are inclined at equal angles to OS, Join SQ, SQ', S'Q, S' Q' and produce S' Q' to W, and let SQ' meet iS'Q in K. Then angle SV Q' = OQ' W- OS'Q' [Euc i .S2. = iSQ'W- iQS'Q' [Props, xin., xxii. = ^S'KQ'. [Euc. I. 32. Similarly SOQ = iSKQ; .-. SOQ = SVQ\ [Euc. I. l.V Prop. XXIII. 1. Given two tangents to an ellipse and one focus, find the locus of the centre. 2. On OQ, OQ', lengths OR, OR' are taken, equal to OS, OS' respective!}-. Prove that RIV is equal to the major axis of the ellipse. 62 ELLIPSE. Proposition XXIV. The locus of the middle points of any system of jjarallel chords of an ellipse is a sti^aight line jmssing through the centre ; and the tangent at either end of the straight line is parallel to the chords. Draw the circle wliose projection is the ellipse. The middle points of the system of parallel chords of the ellipse are the projections of the middle points of a system of parallel chords of the circle. [Props. /3 and 7. In the circle these middle points lie on a straight line cv passing through the centre c. [Euc. ill. 8. And the projection of cv is a straight line CV passing- through the centre (y of the ellipse. [Prop. a. In the circle the tangents at either end of cv are parallel to the chords, because they are all perpendicular to cv. [Euc. III. 3 and 16. Hence in the ellipse the same is true. [Props. 7 and B. Def. The locus of the middle point of a system of parallel chords is called a diameter. Note. The words diameter and axis are frequently used to denote the length of the portion of the diameter or axis intercepted by the curve. Def. The half (QV) of a chord (QQ') ^^hich is bisected by a diameter (OP) is called an ordinate to the diameter. ELLIPSH. i]:i Proposition XXV. Tdvrjents at the ends uf anj chord meet on the diameter vjhich bisects the chord. Let OQ, OQ' be the tangents, join CO, meeting QQ' in I''. Draw the circle whose projection is the ellipse, and let 0, Q, Q\ C, V be the projections of o, q, q, c, v. Join cq, cq. Then oq, oq touch the circle; .'. oq = oq'; .'. angle oc^y = angle ocq\ .'. qv = q'Vj .-. QV=Q'V. [Prop. 8. [Euc. TIL 8(1. [Euc. I. 8. [Euc. I. 4. [Prop. ^. Prop. XXV. 1. Tho taiifient at a yoint P of an ellipse meets the tangent at .1 in Y. Shew tliat CY is parallel to A' P. *2. If CP meets the directrix in Z, ZS is perpendicular to QQ'. 64 ELLIPSE. Proposition XXVI. QV is an ordinate of the diameter CP ; if the tangent at Q meets the diameter CP jiroduced in O, tJien C V . CO = CPl Draw the circle whose projection is the ellipse. Let c, g, 0, p, V be the projections of C, Q, 0, P, V. Join cq and produce qv to meet the circle at q\ [Prop. 8. [Prop. 13. [Euc. III. 8. [Euc. IIL IcS. [Euc. VI. 8. Then oq is a tangent, g'^'' is bisected at v, .'. cvq is a right angle, and cqo is a right angle, .*. cv . CO = cq^, :. cv . CO = cp^, .-. CV . CO = CF\ [Prop. /5. Prop. XXVI. 1. Vll parallel to PQ meets CQ in li. Prove that Pi? is parallel to the tangent at Q. 2. The tangent at any point P of an ellipse meets the eqniconjugate diameters [see page GG] in T and 1". Shew that the triangles TL'l\ TCP are in the ratio CT^ : CT"-. ELLIPSE. 65 Proposition XXVI. (Aliter.) IJraw the tangent at P meeting QO in R. Draw PTf parallel to OQ meeting QV \n W. Join PQ, R W, Then •.* PRQW is a parallelogram, .-. i^ ir bisects PQ, .". RW passes through the centre, .'. by similar triangles GV: CP=CW : CR = CP : GO, .-. GV. GO = GP\ [Prop. 25. What is the corresponding proposition in the parabola? Apply this method of proof to it. This proof is due to the Master of St John's College, Cambridge. C. G. 66 ELLIPSE. Proposition XXVII. If QV bisects chords parallel to CD, then CD bisects chords parallel to CP. O P ^ Draw AQ parallel to CD meeting CP in V; then J.Q is bisected at V. Join A'Q cutting CD in W. Since AQ is bisected in V and A A' in C, .*. J.'Q is parallel to CP. And '. ■ CD is parallel to AQ, and J. A' is bisected in C, .'. A'Q is bisected in W, .-. CD bisects the chord A'Q which is parallel to CP, .-. CD bisects all chords parallel to CP. [Prop. 24. Def. Two diameters which are so related that each bisects chords parallel to the other are called conjugate diameters. N.B. The tangent at P is parallel to Cl> and the tangent at D is parallel to CP. [Prop- 24. Prop. XXVII. 1. To draw the equiconjugate diameters of the ellipse. 2. The focus is the centre of perpendiculars of the triangle formed by two conjugate diameters and the directrix. ELLIPSE. ')/ Proposition XXVIII. Conjugate diameters in the ellipse are the projections o/' diameters in the circle at right angles to one another. Let GP, CD be conjugate diameters. Draw a chord QVQ' parallel to CD and bisected at V. Draw the circle whose projection is the ellipse aiid let D, Q, P, Q', 1", C hv the projections of d, q, p, q, i\ c. cd is parallel to cjq\ and qq' is bisected at v, .'. cv is perpendicular to qq\ .'. cp is perpendicular to cd. [Prop. y. [Prop. y3. [Euc. III. .S. Note. Numerous metrical proi^erties of conjusato iliamcters may l>f deduced from this proposition by the method used in Prop, xxx., e.g. : 1. P'CP, CD are two conjugate diameters, R any other point on the eUipse. Fli, P'li meet CD or CD produced in T, t. Prove CT .Ct = CD-. 2. If CP, CD, CQ, Cli be two pairs of conjugate diameters, and if the tangent at P meet CQ, Cli produced in T, t; then PT. Pt = CD^. D 68 ELLIPSE. Dee. Chords {QP, QP), which join any point (Q) on an ellipse to the extremities of a diameter [PCP') are called supplemental chords. Proposition XXIX. Supplemental chords are parallel to conjugate diameters. Draw the diameters GL, CM parallel to the supplemental chords P'Q, QP cutting them in V and W. Then PV: VQ = PC : CP', [Euc. vi. 2. .-. PV=VQ, .\ 0// bisects all chords parallel to PQ, [Prop. 24. that is parallel to CM. Similarly CM bisects all chords parallel to CL. .'. CL, CM 3ive conjugate diameters. The diagonals of any parallelogram circumscribed to an ellipse are con- jugate diameters. ELLIPSE. 60 Proposition XXX. QV is an ordinate of t/ie diameter PCP', CD the diameter parallel to QV, tJien QY2. py p'v = CD= :CF. Draw the circle whose projection is the ellipse, and let P, V, C, P', Q, D be the projections of j9, v, c, p\ q, d. Since CP, CD are conjugate diameters pcd is a right angle. [Prop. 2.S. But qv is parallel to cd. [Prop. 7. Hence qv is perpendicular to cp, •'• fp-^ = P^' • p'v, [Euc. III. o and o.'). .•. qv' : pv . pv = cd^ : cp', but qi- : cd' = QV : CD\ [Prop. 7. pv . pv : cp"- = PV.P'V: CP\ [Prop. 7. .-. (^1"-': PV.P'V=CB': CP\ On QVoT QV produced is taken a point R, such that VR : VQ = CP: CD. Shew that the locus of R is an elHpse, and tind the position of its axes. 70 ELLIPSE. Proposition XXXI. In the triangles CPN, CDR, CR : PN = CA : CB and CN : DR = CA : CB. Draw the auxiliary circle. Produce NP, RD to meet it in j) and d. Join Cp, Cd and draw the tangents pT, PT to the circle and ellipse respectively, intersecting on the axis. [Prop. 15. Then PT is parallel to CD, [Prop. 24. .•. the triangles TNP, CRD are similar, .-. TN :CR = NP: RD = IS^p : Rd, [Prop. 4. and the angle TNp = the angle CRd, .'. triangles TNp, CRd are similar, [Euc. vi. 6. .*. jjT is parallel to Cd, .•. the angle pCd = angle CjjT = a right angle, therefore the angles NpC, dCR are equal, each being the complement of angle j;CiY, .'. the triangles 2)NC, CRd are equal in all respects, [Euc.i. 2G. .-. pN=CR. But pN :PN=CA'.CB, .-. CR'.PN=CA : CB. Similarly CN: DR = CA: CB. ELLIPSE. Proposition XXXI I. CP-^ + CD'^ = CA'^ + CB^ Draw the auxiliary circle. Produce NP, ED to meet it in j) and d. Join Cp, Cd. Then 1)R' : CN^ = CB' : CA\ [Prop. .SI. and PN' : CR' = CB' : CA'\ [Prop. .SI . .-. J)]r + PN' :CN'-h CR' = r^ : CA\ But CN' + Ci^^ = CX'+pN' = CA\ [Pro]). SI. .-. DR + PN'' = CB\ Now CP^ + CD' = ( 'R + OK' + /)7i" + i^V = CA'' -f C/il Prop. XXXI. If the tan<,'ont at 7' meet tlie major axis in T, ami if (^ be the foot of the perpeiidicuhir from C on the tangent, prove that CQ.QT : C'T'^=CN . PN : CD-. Prove ((/) PG : CD = CB : C.-l ; (/>) Pr; : C/) = C J : C'/i ; (c) PG .Pg = CD\ Prop. XXXII. 1. Find the greatest and least values of the sum of a pair of conjugate diameters. 2. CP, CD are conjugate diameters. If PG, Dll lie the normals at /' and /), prove that PG'- ^ Dll" is constant. 72 ELLIPSE. Proposition XXXIII. The area of the parallelogram formed hy tangents at the extremities of a pair of conjugate diameters is constant. PF . CD = CA . CB. Let QRST be the circumscribing parallelogram, then its sides are parallel to GP or CD. [Prop. 24. Draw the circle, whose projection is the ellipse, and let p, c, d, q, r, &c. be the points whose projections are P, G, I), Q, R, &c. Then j^cd is a right angle, because GP, CD are conjugate to one another, [Prop. 28. qrst circumscribes the circle, [Prop. S. and its sides are parallel to cp or cd, [Prop. y. hence qrst is a square, equal to the square on the diameter and constant in area. Hence QRST is also constant. [Prop. e. Again this parallelogram is equal to 4!PF . CD, but if CP, CD are the axes, the area is 4'CA . CB, .-. PF.CD = CA.CB. ELLIPSE. 70 Proposition XXXTV. If two chords of an ellipse intersect, the recUrnjles con- tained by their segments are as the squares of the paralU^l semi-diameters. Let QOQ\ UOU' be the chords and 6T, Cli the paralkl semi-diameters. Draw the circle whose projection is the ellipse, and let q, 0, q, &c. be the points whose projections are Q, 0, Q\ Szc. In the circle qo . oq = uo . oil, [Euc. ill. -i'). and cp' = cr\ .'. qo . oq' : uo . on = cjf : cr"^, but qo . oq : cp' = QO . OQ' : CF\ [Prop. 7. and uo .OH : cr' = UO .OU' : Cli\ [Prop. 7. .-. QO.OQ' : uo.ou' = rp' : cir. Prop. XXXIII. 1. PCr.Pg = CDK (See Prop. 18.) 2. SP.S'P=CD^. 8. CD.SY=BC.SP. 4. CD is conjugate to (7'. If DQ be drawn parallel to SP, and CQ perpendicular to DQ, prove that CQ is equal to the semi-axis minor. 5. From D tangents are drawn to the circle on the minor axis as diameter. Prove that these tangents are parallel to the focal distances of P. 74 ELLIPSE. Prop. XXXIY. 1. The tangents to an ellipse from an external point are proportional to the parallel semi-diameters. 2. If a circle intersect an ellipse in four points, the chords of inter- section are equally inclined to the axis. 3. If a circle touch an ellipse at the points P and Q, shew that PQ is parallel to one of the axes. 4. Deduce Prop. 3 and Prop, 30 from Prop. 34. 5. If PQ, PQ' are chords equally inclined to the axis, prove that the circle circumscribing PQQ' touches the conic at P. HYPERBOLA. Def. a hyperbola is the locus of a point (P) whose distance from a fixed point {8) bears a constant ratio (e), greater than unity, to its distance (Pi/) from a fixed straight line (XiV), {SP = e. PM). The fixed point {8) is called the focus. The fixed straight line {XM) is called the directrix. The constant ratio (e) is called the eccentricity. HYPERBOLA. 75 Proposition L Construction for j)oints on tlie hjjperhola. The perpendicular on the directrix throufjli the focus is an axis of symmetry. To find the vertices A and A\ From the focus >Si draw SX perpendicular t<> tlic directrix. Divide XS in J., so that SA =e.AX; also in SX produced take A' so that SA' = e.A'X. Then A and A' are points on the curve. Take any point X on the straight line Axi', with centre S and radius e.XX describe a circle, through X draw PXP' perpendicular to AA' and cutting the circle in P and 7*', then P and P' are points on the hyperbola. Draw PM, P\]r perpendicular to the directrix, SP =e. XX = e . PM, SP' = e . XX = e . PJP. Corresponding to any point X on the line AA\ we thus get two points P and P' at equal distances c>n opposite sides of A A'; hence the hyperbola is symmetrical with respect to AA\ or A A' is an axis, and the points A and A' are vertices. Note. It may be proved that the circle intersects tlie i>*rpeii(Iioular NP, when A' is in any part of the axis A A', excej)t the part between A and A', hence the hyperbohi lies entirely outside the lines through A and A' j>er- pendicular to the axis, but it is infinitely extended in both directions (see Appendix). 76 HYPERBOLA. Proposition IT. If the chord PP' intersects the directrix in K, SK bisects the angle betiueeri SP and SP'. Join SP, SP\ SK; produce PS to p, and draw PM, P'M' perpendicidar to the directrix. Then SP = e. PM, and SP' = e . P'M' ; .-. SP : SP' = PM : P'M' = PK : P'K, by similar triangles PKM, P' KM'. Therefore ^/v bisects P'Sp. (Euc. vi. a.) HYPERBOLA. t i Similarly if P and P' are on opposite branches of the hyperbola tSK bisects the angle PSP'. Prove that a st. line cuts the hyperbola in two points only. Prop. I. 1. In any conic, if PR be drawn to the directrix parallel to a fixed straight line, the ratio SP : Pli is constant. 2. If an ellipse, a parabola, and a hyperbola have the same focus and directrix, the ellipse will be entirely on one side of the parabola, and the hyperbola on the other. 3. In any conic a chord through the focus is divided harmonically by the focus and directrix. Prop. II. 1. Prove that a straight line can cut a conic in two points only. 2. In any conic if two fixed points PP' on the curve be joined to a variable point Q, and PQ, P'Q meet the directrix in p,})', the angle pSp' is constant. 78 HYPERBOLA. Proposition III. 7/"PN is the ordinate of a jxyint P on tJie hifperhola, PN' : AN. AX is a constant ratio. Join PA, A'P, and let them, produced if necessary, meet the directrix at K and K'. Join *S'P, 8K, SK\ and produce FS to p. HYPERHOLA. 79 By similar triangles PAX, KAX, PN : AX = KX : AX. By similar triangles PA'N, K'AX, PN : ^'.Y=A"A' : AX; .-. PX' : AX. A'N = KX . K'X : A X . A X. But SK bisects the angle ASp, [Prop. 2. and SK' bisects the angle ASP, [Prop. 2. .'. KSK' is a right angle ; .-. KX . K'X = SX'-, [Euc. VI. s. .-. PX' : AX.A'X=SX' : AX.A'X, which is a constant ratio. Def. Take CB"^ : CA^ in this constant ratio, drawing CB perpendicular to AA'. I. Then A A' is called the transverse axis. II. C is called the centre of the curve. III. CB is called the semi-conjugate axis. So that PX' : AX.A'X^^CB' : CA\ Prop. III. 1. PXF' is a double ordinate of an ellipse. Find tbe locus of tin- intersection of AP and A'P'. 2. In the rectangular hyperbola (page 81) PN-=AX. A'X. 3. PNP' is a double ordinate of a rectangular hyperbola. Prove the angles PAP', PA'P' are supplementary. 4. Tbe tangent at any point P of a circle meets a fixed diameter A /> produced in T. Shew that the straight line through T perpendicular to this diameter will cut A P, BP produced in points which lie upon a certain rectangular hyperbola. 80 HYPERBOLA. Proposition IV. If the diagonals of the rectangle, formed hy perpendiculars through the extremities of the axes AC A', BCB',. he produced indefinitely, and the ordinate NP he produced hoth tvays to meet them in p, p', the rectangle Pp . Pp' = CBl Also the curve continually approaches to each diagonal without actually meeting it, and its distance from it hecomes idtimately less than any finite length. Let parallels to the axes tbroiigli A and B meet in R, and let Pp meet the curve at P'. Then PP', pp are both bisected in N ] :. pP'=pP. Bu t pP .2)R = JSy - NP' ; .-. pP.pP^Np'-NP'. [Prop. 1. [Eiic. II. 5. HYPERBOLA. 81 Now pN"" : CN' = AR' : CA' = CB' : GA\ Agaiu PN' : AN.A'N=CB' : CA\ [Prop. 3. or PN' : CN' - GA' = CB' : CA\ [Euc. ii. 6. Subtracting pN' - PN' : CA' = CB' : CA'; .-. pN'-P^'' = CB''; .'. pP.2^'P = GB\ Since the product pP.p'P is constant, of whicli one factor p'P constantly increases therefore pP constantly diminishes and finally becomes less than any finite quantity. And if Pn be drawn perpendicular to OR the ratio Pn : Pp is constant, therefore Pn continually diminishes and finally becomes less than any finite length. Def. When a curve continually approaches to a fixed straight line without ever actually meeting it, but so that its distance from it becomes ultimately less than any finite length, the line is said to be a rectilinear asymptote to the curve. Def. When the asymptotes of a hyperbola are at right angles the curve is called the Rectangular Hyperbola. In the Rectangular Hyperbola the axes are evidently equal. Hence the curve is sometimes called the Equilateral Hyper- bola. (Note. We shall use the abbreviation u. u. for Rectangular h^'perbola.) Prop. IV. The circle on AA' as diameter cuts the directrices in the same points as the asymptotes. C. G. 6 82 HYPERBOLA. Proposition V. The curve is symmetrical with respect to the conjugate axis, and has a second focus and directrix. Also all chords passing through C are bisected at C. Draw the ordinate PN and take ON' = CN. Since P is on the hyperbola, ON is > CA ; .-. CN' h >CA'', therefore a perpendicular through N' will cut tlie h3'perbola. Let it cut it in P'. Then P'N'' : AN'.A'N' = PN' : AN.A'N [Prop. 3. But A'N' = AN and AN' = A'N; :. AN'. A'N' = AN.A'N; .'. P'N" = PN'', .'. FN' = PN. HYPERBOLA. ^13 Join PP', cutting CB av CB produced in n. Therefore P'nP is parallel to the axis, and therefore per- pendicular to BC, and Pa = P'n. Hence corresponding to any P on the hyperbola, there is another point P' on the hyperbola on the opposite side of CB, such that PP' is bisected at right angles by CB, or the hyperbola is symmetrical with respect to the conjugate axis. If we take CS' equal to CS, and CX' equal to GX, and through X' draw a line perpendicular to AA' , the hyperbola can be described with this line as directrix, 8' as focus, and eccentricity the same as before. Prop. VI. (See page 84.) 1. If an asymptote meets the directrix in E, CE = CA, and CES is a right angle. 2. If Pp be drawn parallel to an asymptote to meet the directrix in j^, Pp = SF. 3. Having given the transverse and conjugate axis, find the focus and directrix. 6—2 84 HYPERBOLA. Propositiox YI. SA=e.AX', CA=e.CX\ CS=e.GA; CA' = CS.CX S ' A' X X A S Because A and A' are points on the hyperbola ; .-. SA=e.AX, [Del SA'-=e.A'X [Def. = e.AX'. By subtraction, AA' — e. XX', :. CA=e.GX (a). By addition, B8' = e. AA\ .-. GS = e.CA (yS). .-. CA'' = CS.CX (7). Note. In this fipure the eccentricity is about 2-2, in the fifrurc of prop. 5 the eccentricity is only I'l, the student should observe the effect of this on the relative positions of S, A, X, and on the general shape of the curve. In this figure CB = 2 . CA ; in the figure of the last proposition GA=2 . CB. HYPERBOLA. 8 o Proposition VII. S'P ~ SP = AA'. Meclianical construction for Injperhula. Draw PMM' perpendicular ou the directrices. Then 8P = e. PM, and ST = e. PM' ; .-. 8'P~SP=.e.MM' = e . XX' = AA'. 86 HYPERBOLA. Proposition YII. {continued). Hence the mechanical construction, S'K is a bar of wood hinged at S', and SPK a string stretched tiofht at P and fastened at S and K. also 8'P + PK = constant, 8P + PK = constant, ;SfT -8P = constant. Prop. VII. 1. The locus of the centre of a circle which touches two fixed circles is an ellipse or hyperbola. 2. Given one focus of an ellipse and two points on the curve, the locus of the other focus is an hyperbola. Note. The figures of this chapter have been drawn by using a wooden cone cut by a plane perpendicular to the base. See prop. 3 of the next chapter. HYPERBOLA. 87 Proposition VIII. B XV^S CS : CA = SA : AX\ [Prop. 6. .-. CS + CA : CA = SA + AX : AX = 8X : AX (1). CS : CA= SA' : A'X, [Prop. 6. .-. CS-CA : CA=SA'^A'X : A'X = SX : .I'.Y (2). Therefore, multiplying (1) and (2) together, CS'-CA' : CA' = SX' : AX. AX = CB' : CA' ; [Prop. 3. .-. CS' - CA' = CB' = AS. A'S, [Euc. ii. 5. Prop. YIII. 1. In the R. H. <-= J2, CS^' = 2AC'- and CS = 2CX. 2. If the asj-mptote meet the directrix in E, and the tangent at the vertex in //, SE = IiC, and SlI is parallel to AE. 88 HYPERBOLA. The latus rectum {LL') is the double ordinate through the focus. Proposition TX. SL.GA= CB\ B But SL' : AS.A'S=CB'' : CA\ AS.A'8=CB\ .-. SL" : CB' = CB' : CA'; .-. SL : CB = CB : CA; .-. SL.GA = GB\ [Prop. 3. [Prop. 8. Prop. IX. 1. Prove this Prop, by means of props. G and 8. 2. In tliGR. H. SL = CA. HYPERBOLA. 89 Proposition X. If the tangent at P meets the directrix in Z, PSZ is a rigid angle. Also tangents at the ends of a focal chord intersect on the directrix. Take a point P' on the hyperbola near to P, and let the chord PP' meet the directrix in K, and produce PS to p. Then KS bisects the ancrlc FSp. [Prop. i^. When P' coincides with P (as in figure 2), so that PFK becomes the tangent PZ, and SK coincides with SZ, P'Sp becomes two right angles ; and PSZ is a right angle. Hence ZSp is a right angle, and Zp is the tangent at p, or the tangents at P and p intersect on the directrix. Prop. X. If ZP, Zp meet latus rectum produced iu 7) and d, proveS'Z) = Sd. 90 HYPERBOLA. Proposition XL If the normal at P intersects the transverse axis in G, SG = e . SP. Draw the tangent PZ, join >S'Z, draw PM perpendicular to the directrix, and join 8M. ZMP and Z8P are right angles ; [Prop. 10. therefore the circle, on ZP as diameter, passes through M and >S'. [Euc. ill. 31. Since ZPG is a right angle, PG touches the circle. [Euc. III. 16. Therefore the angle SPG — angle 8MP in the alternate segment. [Euc. ill. 32. Also angle G8P = angle SPM. [Euc. I. 29. Therefore the triangles SPG, SMP are similar; .-. SG : SP = SP : PM ; .-. SG = e.SP. HYPERBOLA. 91 Proposition XII. The tangent and normal to a Jnjperhola at any point P are i-espectivelf/ the internal and external bisectors of the angle between the focal distances. Let TP be the tangent and PG the normal, meeting the transverse axis in T and G. SG = e.SP, [Prop. 11. and S'G = e . S'P ; .-. SG : S'G = SP : S'P ; therefore PG bisects the angle SPS' externally. [Euc. vi. a. Therefore the complements SP2\ S'PT are equal, and PT bisects the angle SPS' internally. Note. Compare this with prop. 13 of the ellipse. Prop. XII. 1. Given one focus of an hyperbola, one point and the tangent at the point, find the locus of the other focus. 2. If an ellipse and hyperbola have the same foci, they intersect at right angles. 92 HYPERBOl.A. Propositiox XIIL Tlie feet of the perpendiculars (SY, S'Y') from the foci on the tangent at P are on the circle described on AA' as diameter. Also if CE, parallel to the tangent at P, intersects S'P in E, PE = CA. Also SY.S'Y'=CB^ Produce >SfFto meet S'P in W. Join GY. In the triangles YP8, YPW, YP is common, light angles PY8, P FIT are equal, angle FP>Sf= angle FPTF; [Prop. 12. .-. SY= YW, SP = PW; [Euc. i. 26. therefore S'W is parallel to CY\ .-. CY=h{S'W) = I (S'P - PS) [Euc. Yi. 2. [Euc. VI. 4. = iAA' CA- [Prop. 7 therefore Fis on the circle on AA' as diameter. Similarly, F' is on the auxiliary circle. Also YCEP is a parallelogram ; therefore PE=GY=GA. Let Y'S' meet the circle in y and join Yy. Then, YY'y being a right angle, Yy passes through the centre G, [Euc. ill. 31. SY=S'y, [Euc. I. 4. SY.S'T = S'y.S'Y' = AS'.S'A' [Euc. III. 35. = GB'' [Prop. 8. HYPERBOLA. 93 Proposition XIV. If the tangent at P meets the tramverse axis in T, CN . CT = CAl [Prop. 12. [Eiic. VI. A. Draw PMM' perpendicular to the directrices. Join SP, S'P. Then, •.• PT bisects the angle >ST>S" ; .-. ST : S'T = SP : ST = PM : P\M = NX : XX'; .-. ST + S'T : S'T - ST = NX + XX' : XX' ■- XX; .-. 2CS : 2CT=2CX : 2CX ; .-. CX.CT=CS.CX = CA\ [Prop. 6. Prop. XIII. The riders on page 52 are also true for the hyperbola. Prop. XIV. 1. Prove prop. 16 of the ellipse by this method. 2. If Tp be drawn perpendicular to the axis to meet the auxiliary circle in p, prove that Np is a tangent to the circle. 3 . Prove CN.NT = AN . XA ' . 94 HYPERBOLA. Proposition XV. If the tangent at P meets the conjugate axis jiroduced in t, and Pn is the perpendicular from P on the conjugate axis, Cn . Ct = CBl Draw the ordinate PJV. Til en, by similg-r triangles, TN : CT = PN : Ct .-. TN.CX : CN.GT^PN' : Gt.PN) :. TN.GN : CA'=.PN^ : Ct.Gn. [Prop. U. But TN. CN = CN' - CT . CN = CN' - CA' = AN.A'N; .\ AN.A'N : CA' = PN' : Ct.Cn. Therefore, alternately, AN.A'N : PN' = CA' : Ct.Cn. But AN.A'N : PN' = CA' : CB', [Prop. 8. .-. Ct.Cn = CB\ [Prop. 14. [Euc. II. 5. HYPERBOLA. 95 Proposition XVI. //" PF is the perpendicular from P on a line tJironfjli C parallel to the tangent at P, and if the normal at P meets the conjugate axis in g, tlien PF.PG = CB*'^ and PF.Pg^CAl Draw RPN, Prn, perpendiculars on the axes meeting CF in R and ?•, and let the tangent at P meet the axes in T and t. Then since the angles at N and F are right angles, there- fore a circle passes round GXFR. [Euc. ill. 22. Therefore PG.PF= PX . PR [Euc. ill. 35. = Cn.Ct = CB\ [Prop. 15. A(]jain, because the ans^les at F and n are ricrht ans^les, therefore a circle passes rt)und gFr)) : .-. PF. Pg = Pn . Pr [Euc. ill. 86. = CN. GT = CA\ [Prop. 14. Note. It will be seen afterwards that the line CFFi, referred to in the enunciation, is the diameter CD conjupatt.' to CF. 96 HYPERBOLA. Proposition XVIL NG : CN = CB^ : CA'^ and CG = e\ CN. Produce GP to meet the conjugate axis in g. Then NG : CN = PG : Pg [Euc. vi. 2. = PG.PF : Pg.PF = CB' : CA\ [Prop. 16. Again, since NG : GN=CB' : CA'- .-. CN-\-KG : GN=CA'-\-CB' : CA'; .'. CG : CN=CS' : CA' [Prop. 8. = e^ : 1. [Prop. 6. 1. Prove that Prop. XVII. " CG.Cn: Cg. CN=BC^ : A(P. 2. In the r. ii. prove (a) CN = NG, {b)PG = Pg=CP. HYPERBOLA. 97 Proposition XVIII. If from any point on the tangent at P, 01 is drawn perj^endicular to the directrix, and OU perpendicular to SP, then SU = e . 01 (Adams's property). Join 8Z, and draw PM perpendicular to the directrix. Then since the angle ZSP is a right angle, ZS is parallel to OU. .-. SU : SP = ZO : ZP = 01 : MP. .: SU : OI=SP : MP = e : 1. .-. SU = e.OI. If be a point on the tangent, such that OQQ', drawn perpendicular to the transverse axis, meets the curve in Q and Q\ then iSU=SQ and 01P= OQ . OQ'. See ellipse prop. 20, figure 2. C. G. 98 HYPERBOLA. Proposition XIX. To draw a pair of tangents OQ, OQ' to a hyperbola from an external point O. Draw 01 perpendicular to the directrix. With centre S and radius e. 01 describe a circle, and draw OU, OU' tan- gents to it from 0. Draw SZ perpendicular to BU meeting the directrix in Z. Join ZO and produce it to meet SU in Q. Draw QN per- pendicular to the directrix. Then SQ : SU = QZ = QN .-. SQ : QN=SU therefore Q is on the hyperbola. And since QSZ is a right angle, therefore OQ is the tangent to the hyperbola at Q. [Prop. 10. So by drawing SZ' perpendicular to SV, and joining OZ' and producing it to meet SU' in Q\ OQ' is the other tangent. Note. This problem is solved by the principles of Proposition 18, but a construction could also be founded on Propositions 12 or 13. OZ 01; OI=e : 1; HYPERBOLA. 99 Proposition XX. Tangents OQ, OQ' subtend equal or supplementary angles OSQ, OSQ' at the focus S according as Q, Q' are on the same or opposite branches of the hyperbola. Draw 01 perpendicular to the directrix. Join OS, SQ, SQ', and draw OU, OU' perpendiculars on BQ, SQ\ Then SU=e.OI=SU\ [Prop. IS. Therefore the triangles OSU, OSV are equal in all respects. [Euc. I. 26. Therefore the uncAQOSU^ancAeOSU'. Therefore, in fig. 1, angle OSQ = angle OSQ' ; And, in fig. 2, angles OSQ, OSQ' are supplementary angles Note. If lies between the directrices, use the left-hand part of fig. 1. 7—2 100 HYPERBOLA. Prop. XX. 1. The portion of any tangent intercepted between the tangents at the vertices subtends a right angle at either focus. 2. The locus of the centre of the inscribed circle of the triangle SPS' is a straight line. 3. In any conic the chord of contact QQ' is divided harmonically by SO and the directrix. Proposition XXI. OQ, OQ' are inclined at equal or supplementary angles to OS, OS' according as Q, Q' ai^e on opposite or the same branches of the hypei-hola. Case 1. Join 8Q, SQ\ S'Q, S'Q\ and produce Q8 to F, and let SQ' meet S'Q in K. Then, Similarly, angle SOQ = OSW - OQS [Eiic. r. 82. = iQ'S]r-iS'QS [Props. 20, 12. = ISKQ. [Eiic. I. 32. S'OQ' = hS'KQ' ; .. SOQ = S'UQ'. HYPERBOLA. 101 Case 2. K o. SOQ = 1SO^-OSQ-()QS = ISO' -iQSQ'-lSQS' = 1S0''- I SKS\ Again, SVQ' = 180° - OQ'S' - OS'Q' = iSQ'S' - IQS'Q' .-. S0Q = 1S0' -S'OQ'. [Euc. I. 32. [Props. 20,12. [Euc. I. 32. [Euc. I. 32. [Props. 12, 20. [Euc. I. 32. In Case 2 the point lies within one of the two angles between the asymptotes, which contain the two branches of the Iwperbola ; in Case 1 lies within one of the other two angles between the asymp- totes. Also the nature of the proof depends slightly upon whether lies between the directrices or not. For Case 1 in the text the point is between the directrices; in this figure it is not so, and A' consequently lies in S'Q produced. Again, the two positions of 0, given in prop. 20, figure 1, will supply opposite examples of Case 2. 102 HYPERBOLA. Def. a hyperbola which has CB and CA for trans- verse and conjugate axes respectively is called the conjugate hyperbola. Note, The conjugate hyperbola has the same asymptotes as the original hyperbola, because they are diagonals of the same rectangle. [Prop. 4. Proposition XXII. If through any point P ou the curve a line he drawn pai^allel to CA or CB, meeting the asymptotes in p, p', tlie rectangle Pp . Pp' is = to the square on CA or CB respectively. The same is true if P be on the conjugate hyperbola. Case 1. Draw Ppp' parallel to CA, meeting CB in 7^. Then PJS^' : CN'-CA' = CB' : CA'; [Prop. V.. .-. Cn' : Fn' -CA' = CB-. CA\ HYPERBOLA. 103 Also Cv: : pn' = CB' : Ba^ = CB' .-. Pn'-pn' = CA'; or Pjj . Pp = GA\ CA' Case 2. Draw Pj)})' parallel to GB. Then Pp . Pp' = CB\ [Prop. 4. 104 HYPERBOLA. Cases 3 and 4. that Since it has been proved for both axes of the hyperbola Pp . Pp = CA^ or CB"^ respectively, therefore it is also true if P be on the conjugate hyperbola, as in the figures below. Prop. XXIII. QQ' is a chord of a hyperbola parallel to the tangent at P. P/>, Qq, Q'q' are drawn parallel to one asymptote and terminated by the other. Prove Cq.Gq'=Cp'^. HYPERBOLA. 10 .') Proposition XXIII. If tlivoiujli any two points P, Q on the curve or its conjugate two parallel strair/ht lines be draiun to meet the asynijytotes in p, p'; q, (]' respectively, the rectangle Pp . Pp' = Q<1 . Qq'. First let P and Q be on the same branch of the hyperbola. Through P and Q draw Hnes parallel to CB meeting the asymptotes in ii, u'\ w, w'. By similar triangles, Pjo : Pu = Qq : Qiv, and Pp : Pu' = Q(/ : Qw\ Therefore, by multiplying, Pp.Pp' : Pu.Pu' = Qq.Qq' : Qiu.Qw. But Pu . Pu' = CB" = Qw . Qw'; [Prop. 22. .-. Pp . Pp' = Qq . Qq\ The same argument applies whether Q be on the hyper- bola or its conjugate ; both cases are shewn on the figure. Note. Through the centre draw CD parallel to Qq or Pp, meeting the curve or its conjugate at D, then applying this proposition to the points Q and D, Qq.Qq' = DC .DC=CD-. 106 HYPERBOLA. Proposition XXIV. If any straight line cut the curve in Q, Q', and the asymptotes in qq', Qq = Q'q'; And if the tangent rPr' meet the asymptotes in r and r', then Pr = Pr. \ Qq . Qq' = Q'q' • Q'q ; [Prop. 23. .'. Qq. QQ' + Qq . Q'q = Q'q • QQ' + Q'q' • Qq \ '• Qq-QQ' = Q'q'-QQ'; .-. Qq = Q'q'. Let QQ' move parallel to itself until it becomes the tan- gent at P. Since Qq = Q'q always ; .-. Pr = Pr\ Note. QQ' may be on opposite branches of the hyperbola, in this case there is not a tangent to this hyperbola parallel to QQ'. Prop. XXIV. 1. The same is true if qq' be on the conjugate hyperbola. 2. If the normal at P meet the axes in G, g; G, g, r, r' lie on a circle passing through the centre. HYPERBOLA. lo: Proposition XXV. The locus of the middle jyoints of a system of parallel chords is a strai'jJtt line passing through the centre ; And the tangent at either end of the straight line is parallel to the chords. Let QQ', EE' , &c. be a system of parallel chords meetiuo- the asymptotes in q, q'\ e, e! \ &c. Draw CV bisectiug- QQ' in V. Then GV also bisects qq, because Qq = Qq'' [Prop. 24. Therefore, by similar triangles, CV bisects ee\ Therefore it bisects EE' ', because Ee = E'e\ [Prop. 24. Therefore GV bisects all chords parallel to QQ'. Let GV meet the curve in P, and let QQ' move parallel to itself towards P. Then, since QQ' is always bisected by CPV, Q and Q' ultimately coincide with P; therefore the tangent at P is parallel to the system of parallel chords bisected by GP V. 108 HYPERBOLA. Def. a straight line {CP) passing through the middle points of a system of parallel chords is called a diameter. Def. a straight line (QF) drawn from any point on the curve parallel to the tangent at the extremity of the diameter {PGP) is called the ordinate to the diameter. N.B. If the diameter is the transverse axis, the ordinate has the usual meaning. Note. The length of that portion of a diameter, which is intercepted by the hyperbola or its conjugate, is sometimes called the diameter. Proposition XXVI. If one diameter bisects chords jmixdlel to a second, then the second diameter bisects chords imrallel to the first. Let CP bisect QQ' in V and draw CD parallel to QQ. Produce QQ' to meet the asymptotes in (/, q. Through q draw RqJJr'R parallel to CP, meeting the curve in R and R, and the asymptotes in q, r, and CD in U. Then, because Qq = Q'q, therefore qq' is bisected in F; and GV is parallel to qr\ .-. Cr = Cq' ; [Euc. VI. 2. .-. r'U= Uq\ [Euc. VI. 2. and Rq is equal to R'r', .'. R'U=RU; [Prop. 24. therefore CD bisects all chords parallel to CP. [Prop. 25. HYPERBOLA. ior> Proposition XXVI. (Aliter.) If one diameter bisects chords ■parallel to a second, then the second diameter bisects chords pandlel to the first. IV w; Draw AQ parallel to CD, meeting CP in F. Join A'Q cutting CD in TF. Since AQ is bisected in V and A A' in 6'; therefore A'Q is parallel to CP. And because CD is parallel to AQ, therefore A'Q is bisected in W. Therefore CD bisects the chord A'Q parallel to CP. Therefore CD bisects all chords parallel to CP. Def. If two diameters are so related that each bisects chords parallel to the other, they are called conjugate diameters. Note. Of two conjui^jato diameters cue will meet the hyperbola, and the other the conjugate hyperbola. 110 HYPERBOLA. Def. Chords {Ql\ QF) which join any point {Q) on a hyperbola to the extremities of a diameter {POP') are called supplemental chords. Proposition XXVII. Supplemental chords are j^arallel to conjugate diameters. 'V Draw the diameters GL, CM parallel to the supplemental chords P'Q, PQ cutting them in W and V. Then PV : VQ = PC : GP'; [Euc. Vl. 2. .-. PV=VQ; .'. GL bisects PQ, and all other chords parallel to GM. [Prop. 25. Similarly GM bisects all chords parallel to GL ; therefore GL, GM are conjugate diameters. HYPERBOLA. Ill Proposition XXVIII. Tangents to the hyperbola and its conjugate at their inter- sections with conjugate diameters POP', Y)C>T)' form a jxiral- lelogram luhose angular points are on the asymptotes. Also PD is bisected by one asymjytote and is parallel to the other. Draw the tangent rPi-' meeting the asymptotes in r and ?*'. Join CD. Then since CD is conjugate to OP, .'. CD is parallel to rr'. Therefore, by Prop. 23, observing that DC meets both the asymptotes in 0, DC" = Pr . lY = iV; [Prop. 24. .'. DC = Pr and is parallel to it ; .-. rD is parallel to CP ; [Euc. i. 33. .'. rD is the tangent at D. [Prop. 25. Similarly the tangents at D and P' meet on the asymp- totes, and the four tangents form a parallelogram with its angular points on the asymptotes. Join PD, and let rD meet the other asymptote in k. Then rP = Pr', and rZ) = Dk ; .•. PD is parallel to kr, and CPrD is a parallelogram, .•. PD is bisected by the asymptote. For riders see page 113. 112 HYPERBOLA. Proposition XXIX. Straight lines through P and D parallel to the axes form a rectangle with two angular points on one of the asymptotes. Draw Pp parallel to CB, meeting the asymptote in p ; and join j[)i). Let AB, PD intersect the asymptote at k and o» then AB and PD are both bisected by the asymptote, and they are parallel to one another (Prop. 28) ; Hence poP, aKA are similar triangles. .-. Pp : Aa = Po : Ak = PD:AB. [Prop. 28. And angle pPD = angle aAB. Therefore the triangles 2)PD, aAB are similar. [Eiic. VI. (i. Therefore jjZ) is parallel to aB, i.e. to CA. Similarly, if Dd be drawn parallel to CB, Then Pd is parallel to CA. HYPERBOLA. 113 Proposition XXX. CP^ ~ CD-' = CA-^ ~ CB''. Draw the ordinates PiY, DR to the axes and produce them to meet in p, then p lies on the asymptote (Prop. 29). Then Also CB'=pX''-PK'' = cy - cp\ GA'=pR'-DE' = Cp' - CD' ; CA' ~ CB' = CF' - CD\ [Prop. 24. [Euc. I. 47. [Prop. 24. [Euc. I. 47. Prop. XXVIII. In the R. H. prove 1. CP=CD and the asymptotes bisect the angle between any pair of conjugate diameters. 2. CP and CD make complementary angles with the axes. 3. Diameters at right angles are equal. 4. The angle between any two diameters is equal to the angle between their conjugates. 5. The angles subtended by any chord at the extremities of a diameter PP' are equal or supplementary. (). If a R. n. circumscribe a triangle, the locus of the centre is the nine- point circle. C. G. 8 114 HYPERBOLA. Proposition XXXI. If any tangent rPr' to the hyperbola meet the asymptotes in r and x' , the parallelogram CPrD is constant, {or PF . CD = AC . BC). Also the triangle rCr is constant. Draw Aa, Ba parallel to the axes, meeting the asymptote m a. Draw the double ordinate through P meeting the asymp- totes in p, p'. Complete the parallelogram DpPd. Join DP cutting the asymptote in o. Join xiB. Then A DCP : A DpP = Co : op = p'P : Pp. [Euc. VI. 2. Again, A BCA : A DpP = BC : Pp' [Euc. vi. 19. = Pp . Pp' : P/ [Prop. 22. = P2/ : Pp; HYPERliOLA. 115 .-. triande DGP = triangle BGA. .•. parallelogram CPrJJ = parallelogram CAuB, which is constant. Or PF . an = AC . BO. [See fig. of Prop. IG. Also the triangle /•C/''= parallelogram GPrD, for they are, each of them, a (piarter of the parallelogram formed by the tangents at P, D, P', D'. Therefore the triangle rCr' is constant. Prop. XXXI. 1. If Po, Po' be drawn respectively parallel to one asymptote and terminated by the other, Po . Po' = ^CS'^. 2. If the two asymptotes and a point on the curve be given in position, tind the axes and foci. 3. Two tangents to an hyperbola meet the asymptotes in R, r, T, t respectively. Prove lit parallel to rT. 4. In the u. h. if CZ be drawn perpendicular to the tangent at P, prove that CZ.CP:=CA-. 8—2 116 HYPERBOLA. Peopositiox XXXII. QV is an ordinate of the diameter PCP', CD the diameter parallel to QV. Then QV : PV . P' V = CD' : CP'. Let QF meet the asymptotes in q, q. Draw the tangents at P, D, meetmg the asymptotes in ?•, (Prop. 28.) Then CD' = Qq . Qq' [Prop. 23. = qV'-QV'; ... QV'=qV'-Cn\ Also PV.P'V=GV'-CP\ But, by similar triangles, CFr, CVq ; CV'-CF' : CP' = qV'-Fr'' : Pr = qV' -CD" : CD'; .-. PV.FV : CP' = QV' : CD'. Alternando. QV : PV.P'V=CD' : CP'. IntheR. H. QV^=rV.V'V. HYPER P.OLA. 117 Proposition XXXIII. Taufjents at the ends of (iny chord meet on the diameter which bisects the chord. Let QQ', RR be two parallel chords, join RQ, KQ' and produce them to meet in 0. Bisect QQ' in V, and let OF produced meet RR in W. By similar triangles, QV : RW=OV : OW = Q'F : RW, but QV=Q'V, .-. RW=RW. Since VW bisects the parallel chords QQ', RR it is a diameter passing through the centre C. [Prop. 25. Let jR, R move up to and ultimately coincide with Q, Q' '■> then OQR, OQ'R become a pair of tangents at Q, Q , and they still intersect on the diameter CV. In any conic if a diameter meets the directrix in Z, SZ is perpaniicular to the chords bisected by the diameter. 118 HYPERBOLA. Proposition XXXIV. QV is an ordinate of the diameter CP ; if the tangent at Q meets CP in 0, then CV . CO = CPl Draw PU parallel to OQ, and PR parallel to QV, and join PQ. Then PR touches the hyperbola. [Prop. 25. RP UQ is a parallelogram ; therefore R U bisects PQ ; therefore R U passes through the centre C. [Prop. 83. Now therefore CO : CP = CR : RU = CP : CV, CP' = CO.CV [Euc. VI. 2. [Euc. VI. 2. Prop. XXXV. 1. If a R. H. circumscribe a triaugle, it also passes through the ortho- centre. 2. If OR be drawn parallel to an asymptote to meet the curve in R, and OPP' parallel to a fixed line to meet the curve in P, P', the rectangle OP . OP' varies as OR. [See also riders on Prop. 34 of Ellipse. ] HYPERBOLA. 119 Proposition XXXV. If two chords of a hyperbola intersect, the rectanrjles con- tained by tJieir seynients are as the squares of tJie parallel semi-diameters. [Euc. II. 5. [Euc. II. 5. Let the chords POP', QOQ' meet the asymptotes at pp , qq. Bisect PP' at V. Draw kQk' parallel to pp . Then pO . Op' = p V - V\ [Euc. ii. o. PO.OP' = PV'-OV'- .-. pO . Op' - PO . OP' = p V - P V = pP.l'p\ .-. pO . Op -pP . Pp = PU . OF. Similarly, qO . Oq - qQ . Qq' = QO . OQ'. By similar triangles, pO : qO = kQ : qQ, and Op : Oq' = Qk-' : Qq'; .-. pO . Op' : qO . Oq = kQ . Qk' : qQ . Qq = pP.Pp':qQ.Qq';[FT0i^.2S. .-. pO . Op' - pP . Pp : qO . Oq - qQ . Qq = pP.Pp:qQ.Qq'; or PO . OP' : QO . OQ'=pP . Pp' : qQ . Qq = ratio of si^uares of parallel semi-diameters. [Prop. 23. 120 hyperbola. Propositions peculiar to the Rectangular Hyperbola. 2. vw = AN . na; 3. Latus Rectum = AA'. 4. CN = NG. 5. A circle, whose centre is any ]}oint P on the curve and radius PC, intersects the normal on the axes, and the tangent on the asymptotes VQ=VQ = V^ = Pr = Pi^ 6. Conjugate diameters are equal, and the asymptotes bisect the angles between them. 7. Conjugate diameters are inclined to either aocis at angles ivhicJi are comp>lementary. 8. Diameters at right angles to one another are eqnal. 9. The angle between any ttuo diameters is equal to the angle between their conjugates. 10. The angles subtended by any chord at the extremities of a diameter PP' are equal or supplementary. 11. If CZ be drawn j^erjyendicidar to the tangent at P, CZ . CP = CAl 12. If a rectangular hyperbola circumsciHbe a tmangle it passes tlirough the orthocentre. 18. If a rectangidar hyperbola circumscribe a triangle, the locus of its centre is the nine-point circle. CYLINDER AND CONE. If a rectangle revolves round one of its sides, the opposite side traces out a surface, called a right circular cylinder. The length of the rectangle may be considered to be indefinitely extended. The fixed side, about which the rectangle revolves, is called the axis of the cylinder. Def. a right circular cylinder is a surface traced out by a straight line, which moves round the circumference of a circle, and remains always parallel to a fixed straight line, drawn through the centre of the circle, perpendicular to its plane. Def. The fixed straight line is called the axis of the cylinder. Note. The section of a cylinder by a plane parallel to the axis is two generating lines of the cylinder. The section of a cj'linder by a plane perpendicular to the axis is a circle. Def. When a cylinder is cut by a plane, the plane passing through the axis of the cylinder and perpendicular to the cutting plane is called the axial plane. Note. The intersection of the axial plane with the cutting plane is an axis of the curve of section; and its intersection with the cylinder is two generating lines. Def. a sphere inscribed in a cylinder, so as to touch the cylinder in a circle and the cutting plane at a point, is called a focal sphere. 122 CYLINDER AND CONE. Proposition I. The section of a right circular cylinder, hy a plane in- clined to the axis, is an ellipse. Let APA' be the curve of section. Take the axial plane for the plane of the paper, and let it meet the cutting plane in the straight line A' AX and the cylinder in the generating lines KAF, K'F'A' Draw a focal sphere, touching the cylinder in the circle KRK' and the cutting plane at S. Let the planes K'RK, A' PA meet in the straight line XM. CYLIXDEK AND CONE. 128 Throu^^h any point P in the curve A PA' draw a plane F'PFN perpendicular to the axis of the cylinder, meeting the cutting plane in the straight line PN, the axial plane in the straight line FNF' , and the cylinder in the circle FPF'. Through P draw the generating line PR, touching the focal sphere at R ; also draw PM parallel to NX. Suppose SP to be joined. Because the planes APA\ FPF' are both perpendicular to the axial plane, PiV is perpendicular to axial plane (Euc. XI. 19); hence PN is perpendicular to both AA' and FF\ Tangents to a sphere from the same point are equal (p]uc. III. 36); .-. SP = PR = FK, and SA=AK and PM = NX. But FK : NX = AK : AX ; [Euc. vi. 2. .-. SP : PM = SA : AX. Now AK is less than ^A" (Euc. 1. 19), therefore SA : AX is a constant ratio less than unity, and A PA' is an ellipse wliose focus is S and directrix XM. 124 CYLINDER AND CONE. Proposition I. (Second Method.) Let A PA' be the curve of section. Take the axial plane for the plane of the paper, and let it meet the cutting plane in the straight line AA^ and the cylinder in the generating lines KAk, K'A'k'. Draw the two focal spheres touching the cj^linder in the circles KRIC, krk\ and the cutting plane at S and S\ Through any point P on the curve APA' draw a gene- rating line RPr, touching the focal spheres at jR, r. Join PB, PS' which will also touch the focal spheres. Then *SP = Pi^, because they are tangents to a sphere; and S'P = Pr. :. SP + S'P = PR -h Pr = Rr = KL Hence the curve is an ellipse whose foci are S, S' and major axis equal to Kk. (Ellipse, 8.) CYLINDER AND CONE. 125 Proposition I. (Third Method.) Let A PA' be the curve of section. Take the axial plane for the plane of the paper, let it meet the cutting plane in the straight line AA\ and the cylinder in the generating lines AFL, A'F'L. Through any point P in the curve of section draw a plane F'PFX perpendicular to the axis of the cylinder, meeting the cutting plane in the straight line PN, the axial plane in the straight line FXF', and the cylinder in the circle FPF', Draw AL', A'L parallel to KK'. Because the planes KNK' , A PA' are both perpendicular to the axial plane, PJS^ is perpendicular to the axial plane (Euc. IX. 19), hence PX is perpendicular to both FF' and AA'. By similar triangles, AN : XF=AA' : A'L, and A'X : XF' = A'A : AL' ; .'. AX.A'X : XF.XF' = AA'' : A'L.AL'; .'. AX.XA' : PX' = AA" : AL'\ [Euc.iii.35. Hence the section is an ellipse of which A A' is the major axis, and the minor axis is equal to AL\ (Ellipse, 3.) 126 CYLINDER AND CONE. If a right-augled triangle revolves round one side containing the right angle, the hypothenuse traces out a surface called a right circular cone. The length of the hypothenuse may be supposed to be indefinitely extended in both directions. The fixed side, about which the triangle revolves, is called the axis of the cone. Thp angle of the triangle at which the hypothenuse and the fixed side intersect is the i^ertex of the cone. The complete cone when the hypothenuse is indefinitely extended in both directions consists of two equal and similar sheets on opposite sides of the vertex. Def. a right circular cone is a surface traced out by a straight line, which moves round the circumference of a circle, and passes always through a fixed point in a fixed straight line drawn through the centre of the circle, perpen- dicular to its plane. Def. The fixed straight line is called the axis of the cone. Def. The fixed point in the axis is called the vertex of the cone. Note. The section of a cone by a plane passing through the vertex is either a point, or two generating lines of the cone. The section of a cone by a plane, perpendicular to the axis, not through the vertex, is a cii'cle. Def. When a cone is cut by a plane, the plane passing through the axis of the cone and perpendicular to the cutting plane is called the axial plane. Note. The intersection of the axial plane with the cutting plane is an axis of the curve of section: and its intersection with the cone is two gene- rating lines. Def. a sphere inscribed in a cone, so as to touch it in a circle, and the cutting plane at a point, is called a focal sphere. Proposition II. TJie section of a cone by a ^^Icine not passing through the vertex and not perpendicular to the axis satisfies the definition of a conic section (SP = e . PM). CYLINDER AND CONE. O 127 Let AP be the curve of section. Take the axial plane for the plane of the paper, and let it meet the cutting plane in the strait^ht line NAX and the cone in the generating lines OKAF, OK'F'. Draw a focal sphere touching the cone in the circle KRK' and the cutting plane at 8. Let the planes K'RK, PA intersect in the straight line XM. Through any point P in the curve AP draw a plane F'PFN perpendicular to the axis of the cone, meeting the cutting plane in the straight line PN^, the axial plane in the straight line FXF', and the cone in the circle FPF' . Suppose the generating line PRO to be drawn, touching the fiscal sphere at R\ also draw PM parallel to XX. Because the planes AP, FPF' are both perpendicular to the axial plane, PX is perpendicular to the axial plane (Euc. XI. ID); hence PiY is perpendicular to both AX and FF'. Tangents to a sphere IVom the same point are equal (Euc. III. 36). Therefore SP = PR = FK, and SA = AX, and PM = XX. But FK : XX = AK : AX; [Euc. vi. 2. .-. SP : PM=SA : AX: Hence APA' is a conic section, having ^' for focus and XJ/ for directrix. 128 CYLINDER AND COXE. Proposition III. A 'plane section of a cone is an ellipse if its focal axis meets both generating lines in the axial plane on the same sheet of the cone ; it is a parabola if its focal aocis is parallel to one of these two generating lines ; it is a hyperbola if its focal axis meets both these generating lines but on different sheets of the cone. Let the axial plane meet cutting plane in AX, the focal sphere in the circle KK'S, and the cone in the generating lines OKA, OK'. Produce K'K and 8A to meet in X the foot of the directrix. [Euc. I. 16. [Euc. I. 5. [Euc. I. 15. Case 1. Produce ^>Sf to meet OK' in xl' . angle OK'X > angle K'XA'. But angle OK'X = angle OKK' = angle A KX ; .•. angle AKX > angle KXA' or KXA, .: AKAX, [Euc. I. U). .-. >Sf^ > .4.Y, [Euc. III. .SU. and the curve is a hyperbola. CYLINDER AND CONE. 131 Proposition IV. In an elliptic section of a cone tlie major axis is equal to the distance hettveen the focal spheres measured along a gene- rating line of the cone. Let APA' be the curve of section. Take the caxial plane for the plane of the paper and let it meet the cutting plane in the straight line AA' , and the cone in the generating lines KAk, K'A'k'. Draw the two focal spheres touching the cone in the circles KRK', krk' , and the cutting plane at /ij' and S'. Through any point P on the curve APA' draw a gene- rating line RPr, touching the focal spheres at R, r. Join P8, PS', which will also touch the focal spheres. Then SP = PR, because they are tangents to a sphere ; and S'P = Pr. .-. SP + S'P = PR + Pr = Rr = Kk. Hence the curve is an ellipse whose foci are S, S', and its major axis is equal to Kk. (Ellipse, 8.) 9—2 182 CYLINDER AND CONE. Proposition V. In a hyperholic section of a cone, the transverse axis is equal to the distance between the focal spheres, measured along a generating line of the cone. CYLINDER AND CONE. 133 Let A PA' be the curve of section. Take the axial plane for the j^lane of the paper and let it meet the cutting plane in the straight line AA\ and the cone in the generating lines KAk, K'A'k'. Draw the two focal spheres touching the cone in the circles KRK', krk', and the cutting plane at S and S'. Through any point F on the curve APA' draw a gene- rating line RPr, touching the focal spheres at R, r. Join PIS, PS\ which will also touch the focal spheres. Then SP = PR, because they are tangents to a sphere, and S'P = Pr. .-. ST ~ SP = Pr ~ PR = Rj^ = Kk. Hence the curve is a hyperbola, whose foci are S and S\ and its transverse axis is equal to Kk. (Hyperbola, 7.) Props. IV. and V. The auxiliary circle lies on the surface of the sphere, whose diameter is the line joining the centres of the focal spheres. 134 CYLINDER AND CONE. Pkoposition VI. In a ^;a?Y(^o?^c section of a cone, the latns rectum is a third "proportional to the distance of the vertex of the cone from the vertex of the parabola, and the diameter of the cir- cidar section of the cone tlirough the vertex of the parabola. Let AP be the curve of section. Take the axial plane for the plane of the paper, let it meet the cutting plane in the straight line AN, and the cone in the generating lines OAF, OLF'. CYLINDER AND CONE. 135 Throiigli any point P on the curve of section draw a plane F'PFN perpendicular to the axis of the cone, meeting the cutting pLane in the straiglit lino. PN and the axial plane in the straight line FNF' and the cone in the circle FPF\ Draw AL parallel to FF'. Because the planes FPF', APN are both perpendicular to the axial plane, PN is perpendicular to the axial plane (Euc. XL 19), hence PX is perpend irulav to both FF' and AN, Take ^AS a third proportional to OL, LA. By similar triangles AN : NF=OL : LA = LA : 4^AS; .-. ^AS,AN = NF. LA = NF . NF' = PN\ Hence the curve ^P is a parabola, of which the latus rectum is 4^^. (Parabola, 3.) And 4^>S^ is a third proportional to OL, LA. 136 CYLINDER AXD CONE. Proposition VIL /?? an elliptic section of a cone, the minor axis is a mean proportional hetiueen the diameters of the circular sections of the cone passing through the ends of the major axis. Let APA' be the curve of section. Take the axial plane for the plane of the paper, let it meet the cutting plane in the straight \mv A A', and the cone in the generating lines OAFL, OA'F'L'. Through any point P on the curve of section draw a plane F'PFN perpendicular to the axis of the cone, meeting the cutting plane in the straight line PN and the axial plane in the straight line FNF' and the cone in the circle FPF'. Draw AL' , A' L parallel to FF'. CYLINDER AND CONE. 1:3: Because the planes FPF' , APA' are both perpendicular to the axial plane, FX is perpendicular to the axial plane (Euc. XL 19), hence FN is perpendicular to both FF' and A A'. By similar triangles and AN : NF=AA' : A'L, A'N :NF' = AA' : AL'-, AN . A'N : NF . NF' = AA": A'L. AL', AN.NA': PN^ =AA'': A'L . AL' Euc. IIL 35 Hence the section is an ellipse of which A A' is the major axis, and the minor axis is a mean proportional between AL' and A'L. (Ellipse, 8.) 138 CYLINDER AND CONE. Proposition VIII. In a hyperbolic section of a cone, the conjugate aocis is a mean proportional between the diameters of the circular sections of the cone, passing through the vertices of the hyperbola. Let AP \)Q one branch of the curve of section, and A' the vertex of the other branch. Take the axial plane for the plane of the paper, let it meet the cutting plane in the straight line AA' and the cone in the generating lines LOAF, A'OL'F'. Through any point P on the curve of section draw a plane F'PFX perpundicular to the axis of the cone, meeting the cutting plane in the straight line FN, and the axial plane FNF' and the cone in the circle FFF'. Draw AL, A'L parallel to FF\ CYLINDER AND CONE. 139 Because the planes FNF\ A PA' are both perpendicular to the axial plane, PN is perpendicular to the axial plane (Euc. XI. 19), hence PN is perpendicular to both FF' and A A'. By similar triangles AN : NF=AA' : A' L, and A'N : NF'= AA' : AL'; .-. AN . A'N : NF . NF'=AA" : A'L . AL' ; .-. AN . A'N : PN' = A A" : A'L . AL' [Euc. III. 35. Hence the section is a hyperbola, of which A A' is the transverse axis, and the conjugate axis is a mean proportional between AL' and A'L. (Hyperbola, 3.) 140 CYLINDER AND CONE. Proposition IX. The asymptotes of a hyperbolic section of a cone are parallel to the two generating lines, ivhich lie in a jmrallel plane through the vertex of the cone. Take the axial plane for the plane of the paper. Let P be any point on the hyperbola, FN an ordinate, S, S' its foci, A, A' its vertices, C the centre, and .Y the foot of the directrix corresponding to the focus S. CYLINDER AXD CONE. 141 Let OF, OF' be generating lines in the axial plane, and FPF'N a plane perpendicular to the axis. Let the focal sphere touch OF at K, then KX is parallel to FF' (Prop. 2), and >S'^ is equal to AK. [Euc. III. 36. Let Opn be a plane parallel to the cutting plane, meeting the cone in a generating line Op^ the axial plane in On, the plane FPF' in pn. The triangles OnF, AXK are similar because On is parallel to AX, and nF to XK. .: On : OF = AX : AK = AX : AS, :. OF=e . On; but the generating lines OF, Op are equal, .•. Op = e . On. In the figure of Hyperbola, proposition 4, CE' = CA'+AB'' = CA' + CB' = OS' ; .-. CR = CS = eAJA; hence pOn is half angle between asymptotes (Hyperbola, 4), but On is parallel to the transverse axis; therefore Op is parallel to an asymptote. 142 CYLINDER AND CONE. Proposition X. If through any 'point two straight lines he draiun, parallel to two fixed straight lines, to intersect a given cone, the ratio of the rectangles contained hy the segments of the lines is constant for all positions of the point. Let OQQ', ORR' be the two lines drawn through parallel to the two fixed straight lines to meet the cone at QQ\ RR'- Through the vertex V draw VG, VII, parallel to the fixed straight lines; meeting a fixed plane, perpendicular to the axis of the cone at G and H. ^ ORE' and VII are not shown on the figure. CYLINDER AND CONE. 143 First consider only the rectangle OQ . OQ'. Let the fixed piano through G and II meet the plane VQQ' in the straight line GL'L, and the cone in the circle LL'. Again let a plane through 0, parallel to the fixed plane GH, meet the plane VQQ' in OKIC, and the cone in the circle KK'. The triangles OKQ, GLV lie in one plane and their sides are parallel ; .-. OQ : OK=GV : GL. Similarly OQ' : OK' = GV : GL ; .-. OQ . OQ' : OK . OK' = GV : GL . GL'. Now for all positions of 0, GV is constant and the rect- angle GL . GL' is constant [Euc. ill. 30. .'. OQ . 0Q' = \xOK . OK'. Similarly OR.OR' = fMX OM . OM', where X and /a are constant, and J/, M' are the intersections of VR, VR' with the circle KK'. :. OK . OK' = OM . OM' [Euc. iii. 30. .'. OQ . OQ' : OR . OR = \ : /^. 144 Important propositions to he proved hy the reader. PARABOLA. 1. If FOp he a chord of a pg^rahola meeting the axis in 0, and PN, pn or dinates, prove that AN . An = AO"^. (See Prop. 3.) 2. The circle circimiscrihing the triangle formed hy three tangents to a parahola passes through the focus. (See Prop. 13.) 3. If OQ, OQ' are tangents, and OV a diameter, prove that the angle SOV is equal to the angle Q'OS. (See Props. 7, 13.) 4. IfP is the end the diameter luhich hisects a chord QQ', and R the end of another diameter meeting QQ' in M, prove that QiM .MQ'=4^SP . RM. (See Prop. 16.) 5. If the diameter through any ptoint R on the curve meets a chord QQ', and a tangent QT at M and T, prove that TR : RM = QM : MQ'. (See Props. 16, 17 and Proof of 19.) 6. If OP touches a parahola at P, and OQR meets at QR, and the diameter through P meets the chord QR in U, prove that OTP=OQ.OR. (See Prop. 19.) 7. If a circle meets a parahola in four points A, B, C, D, the common chords AB, CI) are equally inclined to the axis of tJte parahola. (See Prop. 19.) 8. If a circle cuts a jxirahola in four points the sutn of the ordinates of these four points is zero. (See Props. 15, 19.) PROPOSITIONS. 145 9. If the normals at three jwiiits P, Q, R meet in a j^oiiH, the sum of the ordinates of P, Q, R is zero, and the circle circumscribing the triangle PQR ixisses tltrough the vertex. (By analytical geometry.) 10. If OQ, OQ' he tiuo tangents to a jmrahola the chord QQ' cuts off from the parabola a segment ivhose area is two- thirds of the triangle Oi^Q'. (See Prop. IG.) CONIC SECTIONS. 1. Ko straight line can meet a conic in more than two points. (Prop. 2.) 2. If a circle meets a conic in four points, the chord joining any tiuo of those points makes the same angle tuith the axis as the chord joining the other two points. (Ellipse 34.) 3. To find luJiere a straight line parallel to the axis meets a conic tvhose focus, directrix, and eccentricity are given. [Cons. Let the line meet directrix in M. \\'^ith centre A' and radius e . SX describe a circle. Join SM meeting this circle in ^, X. Draw SP, SP" parallel to Xp, Xp'. PP' are the required points.] 4. The semi-latus rectum is a Harmonic Mean betiueen the segments of any focal chord 112 + SP ' SP' SL ' SP : SP' = SN : SN' = NX-SX: SX-N'X = SP-SL : SL-SP\ 5. The product of the segments of a focal chord varies as the length of the chord. 6. Rectangles contained by the segments of any two inter- secting chords are proportional to the lengths of the parallel focal chords. (Ellipse 34.) 7. Tangents to an ellipse or hyperbola at right angles to one another intersect on a fixed circle, called the Director Circle. (Ellipse 14.) ^S^' c. G. 10 ^ 146 PROPOSITIONS. 8. Prove PG'.CD^CB: GA and Pg :CD = CA : CB. (Ellipse 18 and 33.) 0. Prove SP .S'P = CD' = PG .Pg. (Ellipse 13 aud 18.) 10. If QQ' be a focal chord, jyarallel to a semi-diameter CD, QQ' . GA = 2GD\ 11. If a diameter of a conic meets the directrix in Z, ZS is perpendicular to the chords bisected bg the diameter. (Ellip.se 11 and 25.) 12. If OQ, OQ' be tangents to a conic and QQ' meets the directrix in K^ OSK is a right angle. (Ellipse 22.) 13. If the tangent at P meet any pair of conjugate dia- meters in T and t, PT . Pt = GD\ (Ellipse 28.) 14. Tlie p)rojection of the normal PG on the focal distance SP is equal to the semi-latus rectum. (Ellipse 12.) 15. If OQ, OQ' are a pair of tangents to an ellipse, and a straight line be draivn from to meet the carve in K, M, and QQ' in L, OKLM is divided liarmonically or 2 11 OL OK OM' (Projections.) 10. If CP, GP' be semi-diameters of a conic at right angles to one another, j^rove that pjj2'^ Trp^ *'^ constant. (Director Circle aud Ellipse 33.) 17. If one straight line ptasses through the pole of a second straight line, p^rove that tite second straight line imsses tJirough the pole of the first. (Projections.) % % PROPOSITIONS. 147 SECTIONS OF A CYLINDER AND CONE. 1. At avy point of a jilane section the tangent makes equal angles with focal distances and the generating line. 2. The semi-minor axis of the section is a mean proi^or- tional between the radii of the focal splieres. 3. For all sections of a cone the latas rectum varies as the j)erpendicular from the vertex of the cone on the i^lane of section. 4. An ellipse of any eccentricity may he cut from a right circular cylinder, and may he projected orthogonally into a circle. 10—2 f PEOBLEMS. PARABOLA. 1. QSq is a focal cliord of a parabola drawn parallel to the tangent at P, PG is a normal. Prove QS . Sq = PG'\ 2. Two parabolas have a common focus, and their axes in the same direction : a straight line is drawn through the focus cutting them in four points. Shew that the tangents at these points form a rectangle of which one diagonal passes through the focus. 3. Given the directrix of a parabola and two points on the curve, find the focus. Also draw a tangent parallel to the straight line joining the given points. 4. PJS^Q is a double ordinate of a parabola and APQ an ecjuilateral triangle ; prove that AN= 3 times the Lat. Rect. 5. In a parabola the external angle between two tangents is half the angle subtended at the focus by their chord of contact. 6. OQ, OQ' are tangents to a parabola, the chord QQ' meets the axis in R, and DM is drawn peri^endicular to the axis, prove that AM = AR. 7. If the normal PG at any point of a parabola be divided so that PQ : Q^r is a constant ratio, prove that the locus of Q is a parabola. 8. Two parabolas have a common directrix, prove that their two common tangents are at right angles to one another. 9. The directrix of a parabola is given and also two tangents : find the focus of the parabola, and the points of contact of the tangents. 10. A chord of a parabola is equal to four times the distance of its middle point from the extremity of the diameter bisecting it ; prove that the chord passes through ^he focus. PROBLEMS. 140 11. If OP, OP' are tangents to a parabola meeting the tangent at ^ in Y and Y\ and PP' cuts the axis in K, prove that KY, KY' are parallel to the tangents OP, OF. (This is true for any diameter, and the tangent at its extremity, not only for the axis.) 12. If PF is a tangent at P to a parabola meeting the tangent at the vertex in F, and a circle on PF as diameter meets the axis in K and K\ prove that PK, PK' produced are normals to the curve. 13. Two chords AB, CD of a parabola are produced to meet in 0, and points E, F are taken in AB, CD so that OE'=OA . OB and OF' = OC . OD, prove that EF is parallel to the axis. 14. If a parabola touches the three sides of a triangle its directrix passes through the orthocentre. 15. If two parabolas are drawn through four given points on a circle, their axes intersect in the centroid of the four points. 16. POQ is an acute angle whose sides are tangents to an ellipse at the ends of a focal chord PQ ; find the two foci. ELLIPSE. 17. If the diagonals of a quadrilateral circumscribing a conic intersect in a focus, they are at right angles to each other. 18. Shew how to draw a pair of conjugate diameters in an ellipse inclined at a given angle to one another. 19. P and Q are corresponding points on an ellipse and its auxiliary circle, >Sf is a focus ; prove that SP = the perpen- dicular from S on the tangent to the circle at Q. 20. The normal at P on an ellipse cuts the minor axis in g ; Pn is the ordinate to that axis. Prove that Cg : Cu = CS' : CB\ 150 PROBLEMS. 21. >S» is a focus of a given conic, and from a fixed point on the axis a perpendicular is drawn to the tangent at any point P on the curve. Prove that the intersection of this perpendicular with SP lies on a fixed circle. 22. Draw a normal from a given point (1) on the axis of a parabola, (2) on the major axis of an ellipse. 23. From any point P on a common tangent to two ellipses, which have a common focus >S', tangents are drawn to the ellipses intersecting another common tangent in Q, R. Prove that the angle QSR is constant. 24. Given an arc of a conic, shew how to determine whether it is part of a parabola, ellipse or liy|>erbola. 25. Given two tangents to an ellipse and one focus, find the locus of the centre. 26. A tangent is drawm to a conic meeting the directrices in Z, M. If >S', H be the foci, and LS, MH intersect in iV, shew that XiY = J/iV^. 27. PQ is a double ordinate of a conic, and the straight line joining P to the foot of the directrix cuts the curve in R. Shew that QR passes through the focus. 28. Two chords AP, BQ in an ellipse are produced to meet each other in ; QG, PD are chords parallel to them crossing each other in R, shew that the triangles A0£, CRD are similar, and AB is parallel to CD. 29. If two conies have a common focus and are so placed that they intersect in two points only, then their common chord passes through the point of intersection of the corresponding directrices. 30. A system of parallelograms is inscribed in an ellipse, with their sides parallel to the equi-conjugate diameters: prove that the sum of the squares on its sides is constant. 31. Prove the following construction for drawing a normal to a conic. Draw the ordinate PN, on the axis mark oti* NK, NL each equal to KP, produce PK, PL to meet the curve again in Q, Q\ bisect QQ in F, then PFis the normal at P. PROBLEMS. 151 32. An ellipse is inscribed in a quadrilateral ABCD, and S is a focus of the ellipse ; shew that the angles ASB and CSB are together equal to BSC and DSA. 83. The perpendiculars from the foci on the normal at any point of an ellipse are to one another as the perpen- diculars from the foci on the tangent at that point. 34. Given two tangents to a conic and its centre : prove that the locus of its foci is a rectangular hyperbola. 35. If FN, the ordinate at the point P of an ellipse, be produced to meet the tangent at the extremity of the latus rectum in Q, prove that QN = SP. 30. An elliptic section of a right cone is projected upon a plane perpendicular to the axis, prove that the focus of the curve of projection is at the point where the axis of the cone meets the plane of projection. 37. If OP, OQ are tangents to an ellipse from a point on the auxiliary circle, and PCP' a diameter of the ellipse, prove that QP' passes through a focus. 38. In any conic if PQ, PQ' are chords equally inclined to the axis, prove that the circle circumscribing PQQ touches the conic at P. 39. If two (juadrilaterals, inscribed in an ellipse, have three sides of one parallel to three sides of the other, their fourth sides will be parallel. Hence shew how to draw a tangent at any point of an ellipse with a parallel ruler. (Projections.) 40. If RP is any tangent to a given ellipse at P and SRP a constant angle, prove that the locus of i^ is a circle. 41. At points Q, Q' on an ellipse OQ, OQ' are tangents, and QG, Q'Cr are normals meeting the axis major at G, G', prove that OQG, OQ'G' are similar triangles. 42. Tangents OQ, OQ' subtend equal angles at the foot of the ordinate through 0. 43. An ellipse touches a triangle at the middle points of its sides, prove the centre of the ellipse is the centre of gravity of the triangle. (Projections.) 152 PROBLEMS. PARABOLA. 44. If AR, SY Eire the perpendiculars from the vertex and focus of the parabola on the tangent, j^rove that SY' = SY.AR-\-SA\ [I. C. S. 1884. 45. P is any point on a parabola, *S'F is drawn per- pendicular to AP meeting the tangent at the vertex in R, prove that AR is one-fourth of PX, the perpendicular from P on the axis. [Clare, 1888. 46. A parabola touches in A', B', C the sides of an equilateral triangle ABC, respectively opposite to A, B, C. Prove that A A', BB\ CC meet in the focus of the parabola. [Trin. 1887. 47. A parabola rolls on an equal parabola, the vertices originally coinciding ; shew that the tangent at the vertex of the rolling parabola always touches a fixed circle. [Trin. 1887. 48. P, Q are two points on a parabola such that circles described about P, Q as centres and passing through the focus S cut orthogonally in S and R. If the line joining Q to the points of intersection of the circles meet the directrix in T and T', shew that the angle TPT is equal to half of RPS. [Pemb. 1887. 49. In the parabola if the angle ASP be equal to four- thirds of a right angle, prove that the ordinate at P and the normal at the extremity of the latus rectum intersect on the axis. [Magd. 1888. 50. Given in position two tangents to a parabola and their points of contact, find the focus and directrix. [Qu. 1888. 51. OPy OQ are two tangents to a parabola at P and Q, aS' is the focus ; if OS meet the circle through OPQ again in T, then >S^ bisects OT. [Qu. 1888. 52. If PG be the normal at P, prove that the tangent from any point on the parabola to a circle, centre G and radius GP, is ecpial to the perj^endicular from that point on the ordinate of P. [Jes. 1888. PROBLEMS. 153 53. 5" is a fixed point on the bisector of the exterior angle A of the triangle ABC; a circle is described upon HA as chord cutting the lines AB, AC in P and Q; prove that PQ envelopes a parabola which has H for focus, and for tangent at the vertex the straight line joining the feet of the perpendiculars from H on AB and AC. [Jes. &c. 1888. 54. Points F, y are taken on the tangent at the vertex of a parabola so that SY . SY' is constant, and the other tangents through Y and Y' meet in Q ; prove that the locus of Q is a circle. [JoH. 1888. 55. A circle is described touching a parabola at a point P and passing through the focus. If K be the point at which it cuts the axis again, and A the vertex of the parabola, shew that AK is equal to three times the abscissa of P. [Sel. 1888. 56. Two points P, Q are taken on a tangent to a parabola equidistant from the focus. Prove that the other tangents drawn from P, Q will meet on the axis. [Pet. 1880. 57. P, Q, R are points on a parabola, the chord PR intersects the diameter through Q in S. The chord PQ intersects the diameter throus^rli R in T. Prove that ST is parallel to the tangent at P. [Clare, 1887. 58. S is the focus and SL the semi-latus rectum of a parabola whose vertex is ^. P and Q are any two points in any line through 0, the point of intersection of the tangent at A and tiie diameter through L. Prove that the chord of contact of the tangents from P intersects the chord of contact of the tangents from Q in the straight line which bisects the angle GAS. [Trin. 1886. 59. Prove that, if P be an external point on the axis of a parabola whose focus is S and vertex A, and the tangent at A cut the circle described on PS as diameter in Q, R, then PQ, PR will touch the parabola. Prove that, if any tangent cut the circle in Q\ R', the remaining tangents from Q', R' to the parabola will intersect on the circle. • [Trix. 1887. 60. A point moves so that the sum of its distances from a giveu point and a given straight line is constant, prove that it describes a parabola and find the length of its latus rectum. [Qu. 1887. 154 PROBLEMS. 61. Give a geometrical construction for the axis of a parabola which passes through the four given points A, B, C, D which are such that AB is parallel to CD. [Jes. 1887. 62. A and P are two fixed jDoints. Parabolas are drawn all having their vertices at -4, and all passing through P. Prove that the points of intersection of the tangent at P with the tangent and normal at A lie on two fixed circles, one of which is double of the other. [JoH. 1887. 63. If PN, PL be perpendiculars from P on the axis and the tangent at the vertex, prove that ZiV^ always touches a parabola. [Pet. 1886. 64. A variable tangent to a parabola intersects two fixed tangents in the points T and T' : shew that the ratio ST : ST is constant. [Tkin. 1886. 65. If QD be drawn perpendicular to the diameter PV of a parabola, then Qn" : QV^ = SA : SP. [Trix. 1886. QQ. Through Y the foot of the perpendicular from the focus S on the tangent to a parabola at P, YK is drawn parallel to the axis of the parabola, meeting the normal PG in 7f, SK is joined. Shew that the triangles SKG and SKP are each of them equal to the triangle SPY. [T. H. 1886. 67. If be a fixed point, MM' a fixed straight line not passing through 0, Q any point m MM', and if on OQ as base an isosceles triangle be described on the side of OQ remote from MM' such that the vertical angle OPQ is always double of the acute angle which OQ makes with MM', shew that the locus of P is a certain parabola. [T. H. 1886. 68. If ABC be a triangle inscribed in a parabola, shew that the sides of ABC are four times as lonor as those of a triangle formed by the intersection of tangents parallel to them. [I. 0. S. 1887. 69. The tangents at Pj, P.^, to the parabola w^hose vertex is A and axis AN^N^ intersect in P, and N^, N^ and N are the feet of the ordinates of P^, P^ and P. Prove that P,N^ : P..y, :: AN : AN^ :: AX^ : AK [1. C. S. 1887. PROBLEMS. 155 70. OQy OQ' are tangents to a parabola, OF a diameter. If OF meet the directrix in K and QQ meet the axis in N, shew that OK = SN\ S being the focus. [I. C. S. 1886. 71. If the tangents at the ends of a focal chord FSQ intersect in D, SD will be a mean proportional between ASiindPQ. [I. as. 1883. 72. Find the locus of the centres of circles described within a given segment of a given circle. [Pet. 1887. 73. FSP\ QSQ\ RSR' are three chords through the focus ^ of a given parabola. Prove that the ratio of the areas of the triangles PQR and P'Q'R' is the same as that of the products of the ordinates of P, Q, R and P', Q', R'. [Pet. 1887. 74. A series of parabolas are drawn to touch two given straight lines, one of them at a given point ; shew that the foci lie on a fixed circle and that the directrices pass through a fixed point. [TiUN. 1887. 75. Two equal parabolas, which have a common axis, have their concavities turned in opposite directions. Prove that the locus of the middle point of a chord of either parabola, which is a tangent to the other, is a parabola of one-third the linear dimonsions of the given ones. [Trin. 1887. 76. The normal at P meets the tangent at the vertex in F and the curve again in f. If the axis of the parabola meets at T and G the tangent and normal at P, shew that PF.Pf=TG\ [T. H. 1888. 77. The normal to a parabola at any point P meets the curve again in Q; T is the pole of the chord PQ, and the line joining T to the focus, >S', meets the line drawn through P perpendicular to SP in the point : prove that TS = SO, and that TOQ is a right angle. [JoH. 1887. 78. F is the middle point of a focal chord QQ' of a parabola, tangents at Q and Q' meet at T; prove that the locus of the intersection of the circle described round the triangle I'QQ' and the line TV is a parabola. [Pet. 1887. 156 PROBLEMS. 79. From any point on a parabola normals are dra\Mi to the curve at P,, P^ ; shew that the chord P^P^ passes through a fixed point. [Clare, 1887. 80. Two equal similarly situated parabolas have a common axis ; a tangent is drawn to one of them meeting the other in P and Q ; prove that the perpendicular distance of Q from the diameter through P is constant and that the area of the segment cut otf by the chord PQ is constant. [Pemb. 1886. 81. Determine the point in a parabola at which tlie normal is equal to a given straight line. [T. H. 1887. 82. If the triangle formed by three tangents to a para- bola be isosceles the line joining the intersection of the equal sides to the focus passes through the point of contact of the opposite side with the parabola. [Cath. 1887. 83. Two parabolas having the same focus cut at right angles. Shew that the line joining their vertices passes through the focus and is equal to the focal radius of their point of intersection; also that the locus of the middle points of this line for different pairs of parabolas througli the same point is a circle. [JoH. 1886. 84. PQ is a chord of a parabola, PT the tangent at P, and a straight line parallel to the axis cuts the tangent in T, the curve in E, and the chord PQ in P; prove that TE : EF :: PF : FQ. [JoH. 1886. 85. If PN be an ordinate and a chord QXQ' be drawn through N cutting the parabola in Q and Q\ then the rect- angle contained by the ordinates of Q and Q' is equal to the square on PX. [Sel. 1887. 86. Two fixed straight lines intersect in A, and P is a fixed point; if a circle be described through A and B cutting these lines in C and P, then CD always touches a certain parabola. [Sel. 1887. 87. The normal chord to a parabola at the point whose ordinate is equal to its abscissa subtends a right angle at the focus. [Pet. 1885. PROBLEMS. 157 88. If a circle passing through the focus of a parabola touches the curve at P and cuts it at L and il/, and the axis at N, prove that LP is equal to MN. [Clare, 18.SG. 89. Give a geometrical construction f(jr the position of the directrix of a parabola whose axis is parallel to a given line, the parabola passing through two given points and touching a given line through one of them. [Clare, 188G. 90. If TP, TQ tangents to a parabola subtend angles at the focus which are constant for all positions of T, prove that the distance between the centres of the circles described about the triangles SPT, STQ will vary as ST\ [Clare, 1886. 91. If PQ be a focal chord of a parabola, and R any point on the diameter through Q : shew that the focal chord PR' parallel to PR = ^^ . [Trin. 1885. 92. Points D, E, F are taken on the sides of a triangle ABC and three confocal parabolas are drawn, one touching BF, FE and EG and the other two the corresponding triads of lines; iSf is the common fijcus and the directrices inter- sect in G, H, K. Prove that the triangles DSG, ESH, FSK are equal to one another. [Trin. 1885. 93. Two parabolas have a common focus : and from a point T external to both tangents TP, TQ are drawn to one and tangents TR, TS to the other. If the angles PTQ, RTS are supplementary, prove that PR, QS are parallel or meet at the focus. If they are parallel, prove that they are also parallel to the common tangent to the parabolas. [Pemb. 1885. 94. From two fixed points A, B perpendiculars AP, BQ are let fall on a variable line ; prove that the envelope of the line is a parabola when the area of the quadrilateral ABQP is constant. [Caius, 1885. 95. The normal at one extremity L of the latus rectum of a parabola meets the curve again in P, the tangent at P cuts the latus rectum produced in M and the axis in T: prove that LM is ^ and XT J times the latus rectum, PN being the perpendicular from P on the axis. [K. 1885. 158 PROBLEMS. 96. A is the vertex, S the focus and P an}^ 2:)oint on a parabola ; PK is the ordinate at P, and the perpendicular to SP drawn through S meets the normal at P in Z ; if LM be the ordinate of L, shew that SM = 2AN. [Qu. 1886. 97. P, Q are any two points on a parabola, R the middle point of the chord joining them, RM is the ordinate of R drawn perpendicular to the axis and RG drawn perpendicular to PQ meets the axis in G ; shew that MG is equal to the semi-latus rectum of the parabola. [Qu. 1886. 98. Prove that the latus rectum is the least focal chord which can be drawn in a parabola. [Cath. 1886. 99. Describe a parabola touching three given straight lines and having its focus in another given line. [Pet. 1861. 100. From >S^ the focus of a parabola a line is drawn parallel to the tangent at a point P meeting the curve in Q ; the diameter at P meets SQ in E. Shew that the locus of ^ is a parabola whose latus rectum is half that of the given one. [Jes. 1861. 101. GR is drawn from the foot of the normal at a point P in a parabola perpendicular to SP cutting the circle de- scribed on SP as diameter in L, LS produced meets the tangent at P in 0, shew that the ratio of OS : OP is in- variable. [Sid. 1861. 102. Parabolas are draw'n passing through two fixed points A and B, and having their axes in a given direction ; find the locus of the foci. [JoH. 1861. 103. A series of parabolas is described having the same tangent at the vertex as a given parabola, and their foci lying on the given parabola. Shew that they intersect in the focus of the given parabola. [Pet. 1861. 104. The tangent at any point P of a parabola meets a fixed circle whose centre is the focus in Q, R. If the other tangents to the parabola which pass through Q, R meet in T, and if the tangents to the circle at QR meet in U, shew that 'fU is parallel to the directrix. [Pet. 1882. 105. At the middle point of a focal chord of a parabola a line is drawn perpendicular to the chord and equal to half the chord ; find the locus of its extremity. [Clake, 1882. PROBLEMS. 159 106. From P, PM is drawn perpendicular to the tangent at the vertex of a parabola and MQ perpendicular to AP ; shew that the locus of Q is a circle. [T. H. 18(S2. 107. Through a fixed point on the axis of a parabola a chord PQ is drawn, and a circle of given radius is described through the feet of the ordinates of P and Q. Shew that the locus of its centre is a circle. [Jes. 1882. 108. If OP, OQ are a pair of tangents to a parabola and PQ cut the axis in li, prove that SR is equal to the distance of from the directrix. [Jes. 1886. 109. A circle cuts a given circle orthogonally and inter- sects a given length on a given straight line ; shew that the locus of its centre is a parabola, and that the envelope of its chord of intersection with the given circle is a conic. [Jes. 1886. 110. PSP' is a focal chord of a parabola. The diameters through P, P' meet the normals at P\ P in F, V re- spectively. Prove that PVV'P' is a parallelogram. [Jes. 1886. 111. AGP is a sector of a circle, centre (7, of which the radius CA is fixed, and a circle is described touching the arc AP externally, and also touching CA and CP both produced; prove that the locus of the centre of this circle is a parabola. [JOH. 1885. 112. If the direction of the axis of a parabola inscribed in a triangle is given prove the following construction for the focus. Through A one of the angular points of the triangle draw AD, perpendicular to the given direction, cutting the circle in D, through D draw DS perpendicular to the opposite side cutting the circle in S ; then S is the focus. [Pet. 1884. 113. P, Q and R are three points on a parabola whose focus is S. Through R are drawn RU and RV, respectivelv parallel to the tangents at P and Q, so as to meet the diameter through Q in U and V. Prove geometrically that RU'' = ^SP.QV. Utilize this result to obtain a geometrical proof of the following : — 160 PROBLEMS. TQ and TR, tangents to a parabola, meet the tangent at Pin X and Y. The tangent at the extremity of the diameter through T meets the tangent at P in 0. Then if S be the focus, SP .QR = 2S0.X F. [Joh. 1 886. 114. Two confocal and coaxial parabolas with the con- cavities in opposite directions are met by any straight line parallel to the axis in P and P' and their common chord QQ' meets PP' in R, «hew that RQ . RQ' : PP' is a constant ratio. [Pet. 1884. 115. The circle circumscribing the triangle formed by three tangents to a parabola passes through the focus : prove that the tangent to this circle at the focus makes with the axis of the parabola an angle equal to the sum of the angles made with the axis by the three tangents to the parabola. [Pet. 1884. 116. PQ is normal at P to a parabola and T is its pole : shew that PS passes through the vertex of the diameter through T. [Pet. 1885. 117. A straight line moves so that two fixed circles always cut off equal chords from it, shew that it always touches a fixed parabola whose focus bisects the line joining the centres of the two circles. [Pet. 1885. 118. If the ordinate at each point of a parabola be pro- duced below the axis until it is equal to the distance of the point from the focus ; prove that the locus of its extremity is another parabola, and that the axes of the curves make with each other an angle equal to half a right angle. [Clare, 1885. 119. Two fixed tangents to a parabola TQ, TR are met by a variable tangent in X and Y. If a chord of the para- bola is drawn parallel to XY and equal to XY, it envelops an equal parabola. [Trin. 1884. 120. A line is drawn througli any point P of a parabola perpendicular to the line joining P to the vextex. This line meets the axis in K, and the normal at P meets the axis in G : prove that GK is equal to half the latus rectum. [Trin. 1884. PROBLEMS. 161 121. Throuf^li any point on a parabola two chords are drawn equally inclined to the tangent there. Shew that their lengths are proportional to the portions of their dia- meters intercepted between them and the curve. [Trin. 1884. 122. PHp is a focal chord of a parabola, and upon P8 and 2^^ ^'^ diameters circles are described ; prove that the length of either of their common tangents is a mean pro- portional between AS and Pp. [Trix. 1885. 123. A straight line PQ cuts two fixed straight lines Ox, Oy which are at right angles, in the points P, Q, and the middle point of PQ lies on a fixed straight line AB. Prove that the straight line PQ is always a tangent to a fixed parabola. [Trix. 1885. 124. If PG the normal at P meet the axis in G; and if GQ be an ordinate erected from G ; prove that the difference between the square on PG and (^G is a constant quantity. [Pemb. 1885. 125. In a central conic if a diameter CT cuts one of its chords QQ' in V, the curve in P and the tangent at Q in Ty then CV. CT=GP^\ deduce the corresponding pro- 230sition for the parabola. 126. If PSQ be a focal chord of a parabola, PG the normal at P, PN the semi-ordinate, and if PN produced meet the diameter passing through Q in H: then HG will be perpendicular to PG. [T. H. 1885. 127. From a point on the directrix of a parabola are drawn two tangents, and tlirough the focus >S' two straight lines parallel to these tangents : the part of the directrix intercepted between these parallels will be bisected at 0. [Chr. 1885. 128. An endless string OPQ is fastened at 0, and two small beads P, Q slide on it; the string is kept stretched; the beads moving so that OP is always equal to OQ and PQ always fixed in direction: shew that the loci of P and Q are arcs of two parabolas with a common focus at 0. [Qu. 1885. c. G. 11 162 PROBLEMS. 129. is a fixed point on a fixed circle ; with any point S on the circle as focns, and the tangent at as directrix, a parabola is described ; shew that the locus of the points of contact of tangents from to the parabola is a circle. [Qu. 1885. 130. Given two tangents to a parabola and their points of contact : construct the curve. [Cath. 1885. 131. From any point on a parabola, chords are drawn making equal angles with the tangent at that point ; shew that they are to one another as the parallel focal chords. [Cath. 1885. 132. C is the centre, and D a fixed point on the circum- ference of a given circle, M is the middle point of any chord BS which is parallel to DC. Prove that CR, OS intersect DM on a certain parabola. [Jes. 1885. 133. The polar of a point with respect to a parabola meets the axis in U, and a straight line through U at right angles to the polar meets OS in R : prove that 0S = 8R. [Jes. 1885. 134. Three parabolas have a common tangent. Prove that the points of intersection of their other pairs of common tangents are collinear. [JoH. 1884. 135. If two tangents be drawn to a parabola, the per- pendicular from the focus on their chord of contact passes through the middle point of their intercept on the tan- gent at the vertex. [JoH. 1884. 136. Pairs of equal parabolas are drawn, having a given point S for focus, one touching a given line AB, the other a given line AG. Prove that the envelope of their common tangents is a parabola whose directrix passes through S, and which touches AB and AC at points in one straight line with>Sf. [JoH. 1884. 137. OXP, OYQ, XRY are three tangents to a ])araboIa (focus S) at the points P, Q, R respectively : find the locus of the remaining intersection of the circles SXP, SVP, as the tangent XY varies its position. [Pet. 1883. PROBLEMS. 163 138. From the vertex uf a parabola lines are drawn parallel to the tangents of the curve : prove that the locus of the points where the}' meet the corresponding normals is a parabola. [Clare, 1S84. 139. If two parabolas have a common focus, the line joining it to the intersection of the directrices is perpen- dicular to the common tangent of the parabolas. [Clare. 1884. 140. Three parabolas are drawn having a common vertex and axis, and their latera recta in geometrical progression : shew that if FQ be the chord of contact of a pair of tangents drawn from a point of the outer to the middle parabola, PQ will touch the inner parabola. [Clare, 1884. 141. If any parabola be described touching the sides of a fixed triangle, the chords of contact will pass each through a fixed point. [Trix. 1884. 142. A circle round the focus of a parabola as centre cuts the tangent at a point P in the directrix, and also at the point T. TM is drawn perpendicular to SP, produced if necessary. Prove that SM is equal to half the latus rectum. [Pemb. 1884. 143. Two tangents OQ, OQ are drawn from an external point to a parabola and a perpendicular on the axis from cuts it in N\ prove that NQ, NQ are equally in- clined to the axis. [Caius, 1884. 144. Two parabolas have the same focus and axis, and the tangent at a point P of one parabola meets the tangent at a point Q of the other perpendicularly at T ', shew that T is equidistant from the diameters through P and Q. [Chr. 1884. 145. A parallelogram circumscribes an ellipse ; shew that the circles, each of which passes through the extremities of a side of the parallelogram and through a focus, are all equal. [Chr. 1884. 146. The portion of the tangent at any point P of a parabola intercepted between the tangents at the extremities of a focal chord subtends a right angle at the point where the diameter through P meets the chord. [Caius, 1883. 11—2 164 PROBLEMS. 147. A line is drawn through a fixed point, and through the point where a line perpendicular to it meets a fixed line a perpendicular to the fixed line is drawn: prove that the locus of the intersection of this and the first line is a para- bola. [Clare, 1883. 148. Any one of a system of parallel lines cuts two fixed parabolas in P, P' and Q, Q respectively ; through P, P' and through Q, Q' lines are drawn parallel to the axis of the para- bola on which they lie ; shew that the angular points of the parallelogram so formed are on a fixed conic. [Chr. 1884. 149. A is the vertex of a pai^abola, P any point on the curve, AP is produced to Q so that PQ =AP\ and through Q a straight line MQL is drawn perpendicular to -^Q meeting the axis in M, if QL be equal to QM shew that the locus of X is a parabola and find the normal at L. [Qu. 1884. 150. If the normal at P meet the axis in G the locus of the centre of the circle drawn round APG is a parabola. [Qu. 1884. 151. Having given three tangents to a parabola and the point of contact of one of them, find the focus and draw the parabola. [Cath. 1884. 152. An isosceles triangle is circumscribed to a parabola; prove that the three sides and the three chords of contact intersect the directrix in five points, such that the distance between any two successive points subtends the same angle at the focus. [Trin. 188G. 153. If PP' be any chord of a parabola perpendicular to the axis and if the diameter throuoh P' meet the tancrent and normal at P in Q and R, then will the middle point of QR lie on a fixed parabola. [Jes. 1884. 154. The tangents at two points P, Q on a parabola intersect in T and the normals at the same points intersect in 0. If TL, OA^ be drawn at right angles to the axis meeting it in L and A, prove that TL .AL = ON' . AS. [Jes. 1884. 155. The tangents to a parabola at Q and P intersect in T, and diameters are drawn trisecting PQ. If one of the tangents at their extremities is perpendicular to TP, then will the triangle PTQ be isosceles. [JoH. 1883. PROBLEMS. 165 156. If the chord PQ of a parabola be normal at P, and if QP produced meet the directrix in R, prove that the angle BTQ is a right angle. [JoH. 1883. 157. From R, the middle point of PG, the normal to a parabola at P, two other normals RC2, RC/ are drawn to the curve. Prove that QS, QfS are equally inclined to the axis. [JuH- 1884. ELLIPSE. 1. The lines AB and AC, at right angles to each other, touch an ellipse whose centre is 0, and cut the circle, with centre and radius OA. a second time in the points B and C respectively. Prove that BC and OA coincide with a pair of conjugate diameters of the ellipse. [I. C. S. 1887. 2. If the normal to an ellipse at a point P meet the axis in G, and PSK be drawn through the fx^us S to meet the diameter conjugate to CP in iT; prove that the ratio of CG to SK will be equal to the eccentricity. [L C. S. 1885. o. Construct an ellipse, having given two points as foci, and a given line as tangent- [L C. S. 1884. 4. Prove that the straight line joining the centre C of an ellipse with the point of intersection of the normals at the ends P, Z) of a pair of conjugate semi-diametere CP, CD is perpendicular to the straight line PD. [1. C. S. 1885. 5. If X, X' are the feet of the directrices of an ellipse corresponding to the foci points. [Qu. 1888. 168 PROBLEMS. 28. If XP, the ordinate at a point P of an ellipse, produced meet the perpendicular from C on the tangent at P in R, shew that the locus of R is an ellipse, and that the tan- gents at P, Q, and R to the given ellipse, the auxiliary circle, and the locus of R all meet in a point. [Cath. 1888. 24. Two circles are drawn touching the ellipse at conju- gate points P and D respectively and each passing through C: shew that their radii are to one another as CP is to CD. [Cath. 1888. 25. A parabola is described passing through the foci of a given ellipse and having for focus some point on the ellipse. Prove that its directrix alwaj^s touches the auxiliary circle of the ellipse. Shew also that the point of intersection of the tangents at the foci of the ellipse lies on a circle. [Jes. &c. 1888. 26. Through a fixed point 0, any chord PQ of a given ellipse is drawn ; an ellipse of given magnitude similar and similarly situated to the given ellipse is drawn through P and Q, prove that the locus of its centre is an ellipse. [Jes. &c. 1888. 27. An ellipse of given magnitude turns about its centre ; prove geometrically that the locus of the pole of any line with respect to it is a circle. [Jes. &c. 1888. 28. Of the tangents at the extremities of the minor axis of an ellipse, one meets a latus rectum in E, and the other the corresponding directrix in F ; prove that EF is a tangent to the ellipse. [Jes. &c. 1888. 29. From P any point on an ellipse a tangent is drawn to the minor auxiliary circle meeting the director circle in Q, jR ; shew that PQ, PR are equal to the focal distances of P. [Jes. &c. 1888. 30. Having given the axes of an ellipse, prove that points on the curve are determined by the following construc- tion. Describe circles on the axes as diameters, and draw a straight line from the centre meeting the circles in P and Q; the straight line through P parallel to the transverse axis, and the straight line through Q parallel to the conjugate axis, intersect each other in a point R of the ellipse. PROBLEMS. 169 Prove also, if a concentric circle be described with radius equal to the sum of the semi-axes, and if the line OPQ meet this circle in F, that VR is the normal to the ellip>e at R. [JOH. 1887. 31. PSQ and PS'R are focal chords of an ellipse ; prove that the tangent at P and the chord QR cut the major axis at equal distances from the centre. [JoH. 1888. •32. In the ellipse BC, AC are the semi-minor and semi-major axes and the rectangle AGBD is completed. If the curve bisect ^D, where /S' is the focus, shew that AC + BG' = 2AC . CS. [Sel. 1888. 38. The centre of an ellipse, a tangent, the length of the major axis and a point on a directrix are given. Shew liow to find the directrices. In what cases will the construc- tion fail ? [Pi:t. 1880. 34. PP' is a diameter of an ellipse, prove that the lines joining the foci to the points Avhere the tangent at P meets the corresponding directrices intersect on the ordinate of P. [Clare, 1887. 35. Two tangents TP and TQ are drawn to an ellipse and any chord TRS is drawn, V being the middle point of the intercepted part; QV meets the ellipse in P' ; prove tliat PP' is parallel to ST. [Trix. 1886. 3(). Two points Q and R are taken on an ellipse having DD' for a diameter and QD and RD' meet in P. Prove that an ellipse, similar and similarly situated to the given one, having D for its centre and passing through P, cuts from PP a chord of w^hich DR is the diameter, and from D'Q a chord of which DQ is the diameter. [Trix. 1886. 37. A tangent at any point P of an ellipse intersects the minor axis in T, and TM is drawn perpendicular to SP produced : shew that the locus of M is a circle. [T. H. 1887. 38. is any external point to an ellipse and OS, OS' are drawn to the foci *S and *S*' cutting the curve at the points P and Q, also SQ and S'P are joined intersecting at the point R; a circle is inscribable in the quadrilateral OPRQ. [T. H. 1883. 170 PROBLEMh). 39. If tangents to an ellipse at points P and P' meet on the auxiliary circle, prove that SP and 8'P' are parallel. [T. H. 1887. 40. If Y and Y' be the feet of the perpendiculars from the foci upon the tangent to an ellipse at P, and PN the ordi- nate of P, shew that PX bisects the angle YNY'. [Mag. 1887. 41. If CP, CD be conjugate semi- diameters of an ellipse, PG the normal at P, CZ the perpendicular from C upon the tangent at P, GM the. line through G parallel to CD and meeting the straight line drawn from P to either focus in M, shew that PM is a fourth proportional to CB, CD, CZ. [Mag. 1887. 42. If P and Q be points on an ellipse whose foci are S and H, the four straight lines SP, SQ, HP, HQ, produced if necessary, are tangents to the same circle. [Qu. 1887. 43. The points of contact of tangents to a series of confocal ellipses from a fixed point on either axis lie on a circle. [Qu. 1887. 44. If Y and Z be the feet of the perpendiculars from the foci on the tangent to an ellipse at P, prove that the tan- gents at Y and ^to the auxiliary circle meet on the ordinate of P, and that the locus of their intersection is an ellipse. [Cath. 1887. 45. The tangents at the points P, P' of an ellipse meet in T, and the normals meet the axis in G, G' respect- ively ; shew that PG, P'G' subtend equal angles at T. [Jes. 1887. 46. Prove that the locus of the focus of a parabola which passes through two fixed points, situated on a diameter of a given circle and equidistant from the centre, and which has a tangent to the circle for directrix, is an ellipse whose foci are the two fixed points. [Jes. 1887. 47. Prove that the tangents draAvn from the extremity of a diameter of an ellipse to the circle described on the axis minor as diameter form with the focal distances of either extremity of the conjugate diameter a parallelogram the difference of whose sides is equal to the semi-axis major. [Jes. 1887. PROBLEMS. 171 48. Inscribe in an ellipse a triangle similar to a given triangle. [Clare, 1883. 49. Two conjugate diameters of an ellipse meet the auxiliary circle in Panel Q. If P' and Q' be the points on the ellipse corresponding to P and Q, prove that the tangents at P' and Q' are at right angles. [Jes. 1887. 50. CA, CB are fixed conjugate diameters and CP, CQ variable conjugate diameters of an ellipse; AP, BQ meet in L ; shew that the locus of X is a similar and similarly situated ellipse. [Jes. 1887. 51. If TP, TP' be two tangents to an ellipse and PG, P'G' the normals at P and P', and if on TP and TP' points Q, Q' be taken so that TQ = TG and TQ' = TG\ shew that QQ = 2PU when U is the middle point of GG\ [JuH. 188G. 52. If a rectangle circumscribes an ellipse, prove that its diagonals are the directions of conjugate diameters. [JOH. 1887. 53. TP and PQ are two tangents to an ellipse, one of whose foci is S. PQ and ST intersect in X and from V, the middle point of PQ, a perpendicular VY is drawn to ST; prove that PV : PX . XQ :: SV : >S'.Y. [JoH. 1887. 54. T, T' lie on CA, CB the semi-axes of an ellipse respectively, and TT' is parallel to AB. Prove that two tangents drawn, one from T, the other from T', to two adjacent quadrants of the ellipse will be jDarallel to conju- gate diameters. [Pet. 1885. DO. If >S'Fis the perpendicular from the focus >S on the tangent to an ellipse at P, prove that SY, CP meet on the directrix. [Pet. I88(i. oQ. PP' is a diameter of an ellipse, the tangents at P and Q are at right angles : prove that the normal to the ellipse at Q bisects the angle PQP'. [Clare, 1880. 57. Pp a chord of an ellipse perpendicular to AC is produced to meet the auxiliary circle in P and p, and the normal at P intersects CP' and Q)' in Q and q: prove that PQ = Pq = CD and PQ = BC [Clare, 1886. 172 PROBLEMS. A tangent to an ellipse at F cuts the major axis in T, and CD is the diameter parallel to PT; prove that TF' + CD' = ST . TH. [Clare, 1886. 59. If P be a point on an ellipse, and the focal distance BF meet the conjugate diameter in E, then the difference of the squares on CF and SE will be constant. [Trin. 1885. 60. Two fixed points, Q and R, and a variable point F are taken on an ellipse ; prove that the locus of the ortho- centre of the triangle FQR is a similar ellipse. [Trtx. 1886. 61. Two ellipses have a common focus and equal major axes; if one ellipse revolves about its focus in its own plane, prove that its chord of intersection with the other ellipse envelopes a conic confocal with this ellipse. [Trix. 1886. 62. From a point R on an ellipse two chords RQ, RQ' are drawn parallel to conjugate diameters CF and CD ; the tangent at R meets QQ' produced in T. Prove that ^ : ^^^'=CF' : CD\ [Trix. 1886. 63. Two concentric ellipses have the same major axis, and their semi-minor axes are CB and Ch ; the ordinate of any point P on the first ellipse meets the second ellipse in p : shew that CF' - CB' : Cp' - Cb' = CA' - CB"" : CA" - Ch\ [Trix. 1886. 64. A series of ellipses is described with equal major axes. The ellipses have one fixed commS, S' two constant equal lines are drawn parallel to SP, PS' where P is a point on the ellipse: prove that the locus of the fourth angular point of the parallelogram having the equal lines as adjacent sides is a circle. [Trix. 1885. 104. >S^ and H are foci of an ellipse and T a point on the major axis produced. A circle is described on >S7/ as diameter. Another circle is described to cut the first at right angles and also to cut the major axis at right angles in T. Shew that the latter circle meets the ellipse upon T's polar with respect to the ellipse. [Pemb. 1883. 105. The normal at a point P of an ellipse meets the axes in G, G'. Shew that if GK is the perpendicular from the centre on the tangent at P, the middle point of CG and 0' the middle point of CG\ then will OB = OK = OP, and O'A' = O'K = O'P. [Trix. 1885. lOG. SY and HY' are perpendiculars from the foci S and H of an ellipse upon a tangent and X and X' are the feet of the corresponding directrices ; prove that XY and X'Y' inter- sect on the minor axis. [Trix. 1885. 107. An ellipse is traced on paper, shew how to find its principal axes. [Trix. 1885. 108. If P be any point on the tangent at A, the ex- tremity of the major axis of an ellipse, and if PjT be the other c. G. 12 178 PROBLEMS. tangent from P to the ellipse, prove that PT is longer than PA. [Pemb. 1885. 109. Two similar and similarly situated ellipses, centres G, C touch one another at a vertex A : through A is drawn a chord, meeting the ellipses in P, Q respectively : PC, QC intersect in R. Find the locus of R. [Pemb. 1884. 110. From any point T on the auxiliary circle of an ellipse tangents are drawn, touching the curve at P and Q. If Pp, Qq be the diameters through these points, shew that Pq> Qp will be focal chords. [Pemb. 1884. 111. The angular points of a triangle are a point on a given ellipse, the centre of the ellipse, and a focus of the ellipse : prove that the locus of the centre of gravity of the triangle is a similar ellipse. [T. H. 1885. 112. If the tangent at any point of an ellipse intersect the tangents at the extremities of the major axis in R and R! , then the circle described on RR' as diameter will pass through the foci. [T. H. 1885. 113. Any two fixed points are taken on the major axis of an ellipse ; through one a line is drawn parallel to S'P, through the other are drawn lines parallel to YS, YS' : prove that the latter meet the former in points which are the extre- mities of a diameter of a fixed circle. [T. H. 1885. 114. PGg normal to the ellipse at P meets the axes in G and g. A circle is described on Gg as diameter and another circle described with P as centre, and cutting the former at right angles, intersects PGg in Q, Q' \ prove that the triangles 8PQ, S'PQ' are similar. [Chr. 1885. 115. From any point Q of a given circle QR is drawn perpendicularly to a fixed tangent and is divided m P so that QP : PR is in a given ratio ; shew that the locus of P is an ellipse. [Qu. 1885. 116. If the diameters through the ends of the latera recta of an ellipse are conjugate diameters, then the line joining the foci subtends a right angle at the ends of "the minor axis. [Qu. 1885. PROBLEMS. 170 117. If the normal at P of an ellipse pass through the extremity of the minor axis then the circle, described on the line joining the foci as diameter, will touch the tangent at P to the ellipse. [Qu. 1885. 118. A circle is drawn touching an ellipse in two points P and Q synmietrically situated with regard to the axis and passing through the focus S, shew that ^'P = >S'Q = latus rectum. [Cath. 1885. 119. Project the following theorem : — If OA and OB be radii of a circle at right angles to each other, and P and Q be points lying respectively on the productions of OA and OB ; then PB and QA will meet on the circle if the rectangle AP . BQ be equal to twice the square on the radius of the circle. [JoH. 1884. 120. CA, CB are the semi-axes of an ellipse. If the rectangle ACBV be con^pleted, and the curve bisect SV, shew that AC + BC = 2AC . CS. [Pet. 1883. 121. Tangents are drawn to an ellipse from any point on the line through the focus perpendicular to the axis: prove that the length intercepted by them on the corresponding directrix is bisected by the axis. [Pet. 1883. 122. PSQ, PHR are focal chords of an ellipse, QT, RT the tangents at Q and R. Shew that PT is the normal at P. [Pet. 1884. 123. TP, TQ are tangents to an ellipse at P and Q; Cp, Cq are the respective parallel semi-diameters ; Tp, PC (pro- duced if necessary) meet in L and Tq, QC in J/; PM, QL are produced to meet in V. Prove that TC Vis a straight line. [Pet. 1884. 124. A circle and an ellipse have a common diameter, from any point on this diameter tangents are drawn to the ellipse and circle, prove that the lines joining the points of contact are parallel to a fixed line. [Clare, 1884. 125. A series of ellipses have a common centre and have two conjugate diameters given in direction and also the sum of the squares of their axes, prove that they all touch four straight lines. [Clare, 1884. 12—2 180 PKOBLEMS. 126. Through the centre of an ellipse whose foci are Sy 8' two constant equal lines are drawn parallel to SP, PS' where P is any point on the ellipse. Prove that the locus of the fourth angular point of the parallelogram, having the equal lines as adjacent sides, is a circle. [Clare, 1884. 127. Through a given point 0, a chord OPQ is drawn to a given ellipse : find the stationary values of the rectangle OP . OD, and distinguish between the maximum and mini- mum values. [Trin. 1883. 128. P, Q, JR, are three points on an ellipse, centre G, RP, RQ meet the diameter AG A' wdiich bisects PQ in K and T. Shew that GN . GT=GA\ [Trin. 1884. 129. The diameter parallel to any focal chord of an ellipse is equal to the chord joining the points on the auxiliary circle which correspond to the extremities of the focal chord. [Trin. 1884. 130. Shew how to draw a focal chord of given length in a given ellipse and prove that if the tw^o chords so drawn be PQ and P'Q', then a circle can be described round PP'QQ. [Trin. 1884. 131. If a triangle can be inscribed in an ellipse with its centre of gravity at the centre of the ellipse the triangle must be the greatest triangle wdiich can be inscribed. [Trin. 1884. 132. If the normal PG to an ellipse pass through B, prove that BG is equal to half the distance between the foci. [Pemb. 1884. 133. If a tangent, its point of contact and one focus of an ellipse be given, find the locus of its centre. [Caius, 1884. 134. On TQ, TQ' a pair of tangents to an ellipse, whose foci are 8 and H, TR, TR are taken equal to T8 and TH respectively ; prove that RR' is equal to the major axis, and that if T8 cut RR in F, TW is equal to TQ. [Caius, 1884. 135. A given straight line moves with one extremity on the circumference of a circle the radius of which is e(|ual to the given line, and with the other extremity on a fixed dia- meter of the circle. Shew that every point of the straight PROBLEMS. 181 line describes an ellipse. Also shew that the sum of the semi-axes of each ellipse is equal to the diameter of the circle. [Mag. 1884. 136. If the tangent at a point P of an ellipse meet the tangent at the vertex ^ in jT and >S'' be the focus further from A, tlien TA is equal to the perpendicular from Z'on S'P. [Qu. 1884. 137. If CY, CZ be drawn perpendicular to the tangents to an ellipse at F and D conjugate points, and D' be the opposite end of the diameter CD, shew that PU is the diameter of the circle described round the triangle YCZ. [Qu. 1884. 138. Having given the auxiliary circle of an ellipse and a tangent to the ellipse touching the ellipse at a given point, find the foci of the ellipse. [Cath. 1884. 139. If AA' is the transverse axis of an ellipse, and if Y, Y' are the feet of the perpendiculars let fall from the foci on the tangent at any point of the curve, prove that the locus of the point of intersection of A Y and A' Y' is an ellipse. [Trix. 1885. 140. The perpendicular from C on QQ meets the auxi- liary circle in R ; through G a line is drawn parallel to PR meeting a perpendicular to QQ' through Fin 0. Prove that, if an ellipse be described through Q and Q' with as centre and major axis equal to that of the given ellipse, it will have its minor axis equal to BCD'. [Trin. 1886. 141. Two tangents TP and TQ are drawn to an ellipse and any chord TRS is drawn, V being the middle point of the intercepted part; QF meets the ellipse in F\ prove that PP' is parallel to ST. [Trix. 1886. 142. Two points Q and R are taken on an ellipse having DD' for a diameter, and QD and RD' meet in P. Prove that an ellipse, similar and similarly situated to the given one, having D for its centre, and passing through P, cuts from UP a chord of which DR is the diameter, and from UQ a chord of which DQ is the diameter. [Trix. 1886. 143. Through the foci S, H of an ellipse two lines PSP', QHQ' are drawn meeting two tangents PQ, P'Q' and such 182 PROBLEMS. that PP\ QQ' are bisected in S and H respectively. Shew that a circle can be described about the quadrilateral PQQ'F. [Jes. 1884. 144. In the ellipse if the perpendiculars from G and C on CP and the tangent at P meet in H, and the circle on CH as diameter meet the tangent at P in L, prove that CL is equal to the tangent drawn from P to the circle described on the axis minor as diameter. [Jes. 1884. 145. The locus of the intersection of tangents to an ellipse at right angles is a circle. [Jes. 1884. If the tangent at P cut this circle in T, prove that TP subtends at the foci angles which are complementary. 146. A circle passing through the foci of an ellipse inter- sects the curve at P and Q on opposite sides of the axis. Prove that the sum of the squares of the perpendiculars from the centre on the tangents at P and Q is equal to the square on^a [JoH. 1883. 147. From the foci S, H, SO, HO' are drawn perpen- dicular to SP, HP to meet the normal at P in 0, 0'. Shew that 00' is bisected by the minor axis. [Pet. 1883. HYPERBOLA. 1. Give in magnitude and position the tw^o axes AC A', BOB' of a hyperbola, construct geometrically a pair of con- jugate diameters POP', BCD', which shall contain a given angle. [I. C. S. 1886. 2. A straight line cuts a pair of conjugate diameters of a hyperbola in P and D, and a second pair in P' and D' ; if be the middle point of the line intercepted between the asymptotes, prove that OP .0B= OP' . on. [I. C. 8. 1886. 3. Given one focus, a tangent, and the length of the minor axis a hyperbola, shew that the locus of the centre is a straight line. [I. C. S. 1885. PROBLEMS. 188 4. If two tangents of a hyperbola intersect on one branch of the conjugate hyperbola, prove that their chord of contact touches the other branch. [I. C. S. 1885. 5. Through N the foot of the ordinate of a point P on a hyperbola draw A^Q parallel to J. P to meet CP in Q. Prove that AQ is parallel to the tangent at P. [I. C. S. 1884. G. Two angular points of an equilateral triangle are respectively the centre and one focus of a hyperbola, and one side of the triangle is an asymptote. Find where the other two sides are cut by the curve. [I. C. S. 1883. 7. If two sides of a triangle are fixed in direction and the third passes through a fixed point, the locus of the centres of the circles circumscribing the triangle will be a hyperbola. [I. C. S. 1883. 8. A circle is described having for diameter a chord of a rectanglar hyperbola with its ends on different branches. Prove that the perpendiculars drawn to this chord from the other points of intersection of the circle and hyperbola are tangents to the hyperbola. [Pet. 1887. 9. Given in position the asymptotes and one tangent to a hyperbola, shew how to construct the curve. [Pet. 1887. 10. A circle and a rectangular hyperbola intersect in four points which lie on a given parabola ; prove that an axis of the hyperbola is parallel to the axis of the parabola; and shew that whatever curve the centre of the hyperbola (or circle) describes, the centre of the circle (or hyperbola) will describe an equal curve, the two centres moving over their respective curves in opposite directions. [Pet. 1887. 11. A parabola and rectangular hyperbola, one of w^hose asymptotes is the axis of the parabola, each circumscribe the triangle PQR whose sides cut the axis of the parabola in p, q, i\ respectively. If A be the vertex of the parabola, and PN the ordinate of P, prove that Aq -\-Ar = AN. [Pet. Pemb. &c. 1888. 184 PROBLEMS. 12. With each pair of three given points as foci, a hyperbola is drawn passing through the third point : shew that the three hyperbolas thus drawn intersect in a point. [Trix. 1888. 13. Shew that all the conies which pass through the three vertices of a triangle and the intersection of its three perpendiculars are equilateral hyperbolas : and determine the locus of the centre of these hyperbolas. [LoND. 1st B.A. Hon. 1872. 14. Two points P, Q are taken on a hyperbola so that the tangent at P and a parallel through Q to one asymptote intersect on the other asymptote ; shew that the tangent at Q and a parallel through P to the second asymptote intersect on the first asymptote. [Trix. 1888. 15. Given a hyperbola traced on paper, how would you find its transverse and conjugate axes and its asymptotes ? [T. H. 1888. 16. Having given the asymptotes of a hyperbola and a point on the curve, find the foci, directrices, and vertices. [C. C. C. 1888. 17. (7 is the centre of a rectangular hyperbola, a straight line LQ is drawn parallel to one asymptote CM meeting the other in L, and the angle QCM is bisected by a straight line which meets the hyperbola in P; shew that CQ is proportional to CP'\ Q being any point on the line LQ. [Cath. 1888. 18. The perpendiculars drawn from the foci of a rect- angular hyperbola on the tangent at any point P meet the curve in points K, Z, M and N. Prove that KLMN is a parallelogram two of whose sides are at right angles to the diameter through P. [Jes. &c. 1888. 19. One asymptote and three points of a h}^erbola being given, construct the other asymptote. [Jes. &c. 1888. 20. If P be any point of a liyperbola and AA' its transverse axis, and if. 17^ and AP meet a directrix in E and F, prove that EF subtends a right angle at the corresponding focus. [JoH. 1888. PROBLEMS. 185 21. With two sides of a square as asymptotes, and the opposite point as focus, a rectanguhir hyperbola is described ; shew that it bisects the other sides. [JoH. 18(S8. 22. An ellipse is drawn having its axes, major and minor, coincident in direction and magnitude with those of a hyperbola : from any point T on either asympt(jte, tangents TQ, TQ' are drawn to the ellipse : prove that the circle described round TQQ' passes through the centre of the hyperbola. [Clare, 1887. 23. ABCD is a rectangle. Two equilateral hyper- bolas having their asymptotes parallel to the sides of the rectangle pass through A and 0, and B and Z), respectively. Prove that the polar of the centre of one hyperbola with respect to the other coincides with the polar of the centre of the latter with respect to the former. [Trin. 1886. 24. P is a point in the plane of a triangle ABC, such that the perpendiculars from A, B, C upon PB, PC, PA respectively meet in a point. Shew that the locus of P is a hyperbola circumscribing the triangle ABC and passing through the points of intersection of the perpendiculars let fall from A, B, C upon the opposite sides of the triangle with the straight lines drawn from B, C, A respective! v perpen- dicular to BA, CB, AC. [Trix. 1886. 25. Prove that the parallel focal chords of conjugate hyperbolas are to one another as the eccentricities of the hyperbolas. [Trin. 1887. 26. Find the locus of the intersection of the tangent with a straiofht line drawn from the focus makino- a fixed angle with the tangent. [Trin. 1887. 27. P is a point on a hyperbolic branch whose vertex is A, LPL is the tangent at P terminated by the asymp- totes, and MP AM' is a straight line terminated by lines drawn through the further vertex parallel to the asymj^totes ; shew that LM and L'M' are parallel. [Mag. 1887. 28. If P and Q be any two points on a rectangular hyperbola, G the intersection of the axes, P T the tangent at P, QM and QX the perpendiculars from Q upon CP and PT respectively, shew that CM and CX are equal. [Mag. 1887. 186 PROBLEMS. 29. If P be any point of a hyperbola whose foci are S and H, and if the tangent at P meet an asymptote in T, the angle between tliat asymptote and UP is double the angle >STP. [K. 1886. 30. If a tangent at P meets the asymptotes in L and M the locus of the centre of tlie circle circumscribing the triangle LCM is a hyperbola having its asymptotes at right angles to the original ones. [Qu. 1887. 31. Ox, Oy are any two fixed straight lines ; A lies on Ox and B on Oy and OA — OB. Through A, B, any two jDarallel lines AM, BN are drawn meeting Oy and Ox respec- tively in M and N \ shew that the locus of the middle point of MN is a hyperbola. [Cath. 1887. 32. A circle which passes through two fixed points S, S\ cuts two fixed straight lines, which are perpendicular to SS' and equidistant from its middle point, in the points P, Q, and P\ Q\ Shew that if PP' be not parallel to SS', it will touch a fixed conic whose foci are >S', S'. [Jes. &c. 1887. 33. A rectangular hyperbola is drawn passing through two fixed points P, Q on a fixed conic, and having an asymptote parallel to a given straight line : shew that if it cuts the given conic again in M and S, the straight lines PP and QS intersect on a fixed conic. [Jes. 1887. 34. OX, OY are fixed straight lines; ^ is a fixed point on OX and P a variable point on OF; PM is drawn perpendicular to AX and Q taken on PM so that AQ = PM ; find the locus of Q. [Jes. 1887. 35. P is any point on a circle of which AB is b> fixed diameter. Through B a line is drawn to meet AP produced in Q so that BP, BQ make equal angles with AB. Find the locus of Q. [Jes. 1887. 36. If a triangle ABC be inscribed in a rectangular hyperbola, prove that its orthocentre P lies on the hyperbola. If through P chords Pxi', PB\ PC be drawn parallel to the sides of the triangle, prove that A A', BB', CC ai'e parallel. [JoH. 1886. PROBLEMS. 187 .S7. A and C are points on opposite branches of a rect- angular hyperbola, and the circle described on AC as diameter meets the curve again in B and I). Prove that the distances of any point on the hyperbola from the sides of the quadri- lateral are proportionals. [JoH. 1886. 38. The base AA' of a trianii^le is fixed in mat^nitude and position: prove that if the ditierence of the base angles is a right angle, the locus of the vertex is a rectangular hyperbola. If FN is the perpendicular on AA' and N'Q, NQ' the tangents from iV to the circle on AA' as diameter, prove that PQ passes through A' and FQ' through A ; and also, if QQ' intersect AA' in M, that FM is the tangent at F. [JuH. 1887. 39. If a family of rectangular hyperbolas be described about a triangle, their centres will all lie on the nine-point circle. If the triangle be right-angled, all the hyperbolas will have a common tangent at the right angle. [Pet. 1886. 40. Prove geometrically that the locus of points on a system of confocal ellipses where the tangents are parallel to a given line is an equilateral hyperbola. [Clare, 1886. 41. If the conjugate diameters FCp, DCd of an ellipse be the asymptotes of a hyperbola, QQ' one of the common chords, Q'R, QR chords of the ellipse parallel respectively to CD and CF, prove that Q'R' : QR :: CD : CF. [Clare, 1886. 42. Prove that the common chords of a hyperbola and circle may be grouped in ]iairs which meet the asymp- totes Jn concyclic points ; and that these circles are all concentric with the original circle. [Trin. 1886. 43. Having given, in a triangle, its base and the differ- ence of its base angles, prove that the locus of the vertex is a rectangular hyperbola. When is the base of the triangle the transverse axis ? [Caius, 1885. 44. If two concentric rectangular hyperbolas have a com- mon tangent the anMe between their transverse axes will be half the ano;le between the straioht lines from the centre to the points of contact. [T. H. 1886. 188 PROBLEMS. 45. In a hyperbola, supposing the two asymptotes and one point of the curve to be given in position, find the posi- tion of the vertices. [T. H. 1886. 46. Four tangents to a hyperbola form a rectangle. If one side AB of the rectangle cut a directrix of the hyperbola in X and >Si be the corresponding focus, sliew that the tri- angles XSA, XSB are similar. [Chr. & E. 1885. 47. In the rectangular hyperbola, the angle between a chord PQ and a tangent at P is equal to the angle subtended by the chord PQ at the other extremity of the diameter through P. 48. Two rectangular hyperbolas touch one another in P and intersect in P and S. Prove that the circle on RS as diameter passes through P and the extremities of the two diameters through P. [Chr. & E. 1885. 49. If an equilateral triangle be inscribed in a rect- angular hyperbola, find the locus of the centre of its circum- scribing circle. [Q^^- 1886. 50. In the rectangular hyperbola, prove that the portion of the normal at any point intercepted between the point and the axis, is equal to that semi-diameter of the conjugate hyperbola which is perpendicular to the normal. [JoH. 1861. 51. Parabolas are drawn passing through two fixed points A and B, and with their axes parallel to a given straight line ; if a tangent be drawn at right angles to AB, prove that the locus of its point of contact is a hyperbola. [JoH. 1861. 52. A straight line moves between two straight lines at right angles to each other so as to subtend a right angle and a half at a fixed point on the bisector of the right angle ; prove that it always touches a rectangular hyperbola. [JoH. 1861. 58. Prove that a rectangular hyperbola, confocal to a given elli})se, intersects it at the extremities of its equi-conjugate diameters. [Pet. 1861. 54. If a parabola be described with any point on a hyperbola for locus, and passing through one of the foci of PROBLEMS. 189 the hyperbola, shew that its axis will be parallel to one of the asymptotes. [Pet. 1882. 55. The tangent to a parabola at P meets the tangent at the vertex in Y. The ordinate PN is produced to R so that RN = PY. Shew that the locus of i^ is a rectangular hyperbola. [Jes. 1882. 56. A and B are fixed points on a given circle, and CD is any chord of given length. If CD be drawn parallel to AB, and li AE.BD meet in 0, the locus of is a rectangular hyperbola. [Jes. 1882. 57. Given the auxiliary circle of a hyperbola and a point on the curve, shew that the locus of the foci is an hvperbola. [Jes. 188G. 58. Shew that the locus of the intersection of two equal circles which touch two given parallel straight lines at given points A and B and whose centres are on the same side o^ AB is a hyperbola,. [Jes. 188G. 59. Shew that the angle between two tangents to a rectangular hyperbola is equal or supplementary to the angle which their chord of contact subtends at the centre, and that the bisectors of these angles meet on the chord of contact. [Jes. 1886. 60. The tangent at a point P of a rectangular hyper- bola meets the asymptotes in K and L, and the normal at P meets the axis in G ; find the centre of the circle circum- scribing the quadrilateral CKGL. [JoH. 1885. 61. Two hyperbolas have the same transverse axis and a line perpendicular to it meets them in points P and P'. Prove that the tangents at P and P' meet on the transverse axis. [Pet. 1884. 62. A tangent to a hyperbola at a point P meets an asymptote in T. A lino R'PR is drawn parallel to this asymp- tote, to meet a directrix in R' and the line ST in R, where *S^ is the focus corresponding to the directrix ; prove that R'P^RP. [Clare, 1885. 63. Shew that if the tangent at a point P of a hyper- bola meet an asymptote in T, the angle between CT and HP will be double the angle STP ; where C is the centre, and S and H the foci of the curve. [Trin. 1884. 190 PROBLEMS. 64. Shew that if CP, CD be conjugate semi-diameters of a, hyperbola whose foci are >S' and H, then the distance of i) from a line drawn through C parallel to HP will be equal to the semi-minor axis. [Trin. 1885. Qo. The tangent to a hyperbola at a point P meets the asymptotes in Q, q ; QM, qm are the ordinates of Q, q, and CT the perpendicular from the centre on the tangent at P. If TM, Tm meet the normal at P in /i, L respectively, shew that OKgL is a rhombus. [Pemb. 1885. ^Q. Defining the hyperbola to be the envelope of the line which cuts off from two fixed lines a triangle of constant area, prove that the hyperbola has two asymptotes and that the line touches the curve at its middle jDoint. [G. & C. 1885. 67. Prove that the angle between the tangents at a point of intersection of two concentric rectangular hyperbolas is double of the angle between their transverse axes. [T. H. 1885. 68. Let PQ be any diameter of a rectangular hyperbola and let a circle be described with centre P and radius PQ, then if ^, 5, C be the other points in which the circle cuts the hyperbola, the triangle ABC is equilateral. [K. 1884. 69. A circle meets a given rectangular hyperbola in A, A', P, P\ prove that the tangents to the hyperbola at P, P' intersect in a point lying on the diameter at right angles to A A'. [Chr. 1885. 70. >S is the focus of a parabola whose vertex is A, and SA meets the directrix in X ; SXH is an angle of 60° and SII is perpendicular to SX, shew that a hyperbola may bo described with >S' and // as foci touching the parabola in a point P whose focal distance is equal to the latus rectum. [Qu. 1885. 71. Through a given point P any straight line is drawn meeting two fixed straight lines in P' and Q' ; a point Q is taken on P'Pi/ so that QQ' = PP' ; shew that the locus of Q is a hyperbola. [Catii. 1885. PROBLEMS. 191 72. The tangent and normal at any point of a hyper- bola intersect the asymptotes and axes respectively in four points which lie on a circle passing through the centre of the hyperbola, and the radius of this circle varies inversely as the perpendicular from the centre upon the tangent. [JOH. 1884. 73. If the asymptotes of a hyperbola be inclined to each other at an angle equal to half a right angle, find (an; prove that the axis of the parabola is parallel to one of the lines joining tlie extremities of the diameters of the conic which are parallel to AB and OB. [JoH. 1861. 33. The tangents at two points P, Q of a conic meet in 0, and from are drawn two straight lines cutting the PROBLEMS. ' 197 conic and making equal angles with the transverse axis. If they meet PQ in ^1/, N, and the middle points of the chords be R, S, shew that BMNS lie on a circle. [Pet. 1882. 34. Two conies have their directrices parallel, and the same focus aS' : if any straight line through S meet the two conies in F and Q, find the locus of the middle point of PQ. [Chr. 1882. 35. A, B, C are any three fixed points ; through A any straight line is drawn which cuts a given conic in the points P, Q. Shew that the locus of the intersection of PB and QC is a conic. [Jes. 1886. 3G. is a fixed point, and P any point on a given straight line. PQ is taken along the line always in a constant ratio to OP. Prove that the line joining P to the middle point of (JQ always touches a conic whose focus is 0. [Jes. 1886. 37. Prove that if an ellipse and a hyperbola are confocal they intersect each other at right angles, and that the asymptotes of the hyperbola pass through the points on the auxiliary circle of the ellipse which correspond to the points of intersection. [JuH. 1886. 38. A line AB is drawn from a fixed point A to meet a fixed circle in B: through B a line BC is drawn perpen- dicular to AB, to meet a concentric circle in C. Shew that a line through C parallel to ^i^ touches a conic. [Pet. 1884. 39. Two tangents are drawn from a j^oint on the direc- trix to a central conic, and the points of contact joined. Shew that the locus of the orthocentre of the triangle thus formed is a conic similar to the given one. [Pet. 1884. 40. A fixed straight line meets one of a system of confocal conies in two points. Prove that the locus of the point where the normals at these points intersect is a straight line. [Pet. 1884. 41. With any point on the directrix of a given parabola as focus and the focus of the parabola as the other focus, an ellipse or hyperbola is described, shew that the tangents and normals at its points of intersection with the directrix are also tangents to the parabola. [Pkt. 1884. 198 PROBLEMS. 42. A fixed chord PQ of a conic meets any diameter in N, and the ordinate to this diameter through N meets the tangents at P and Q in H, K. Prove that HK is bi- sected at N. [Caius, 1883. 43. If any two chords PQ, PQ' be drawn through a point P of a conic and perpendiculars to the chord through Q and Q' meet the normal at P in N, N' respectively, shew that PN, PN' are to one another as the squares of the diameters of the conic parallel to PQ, PQ'. [Pet. 1885. 44. If A, B,C, D are four points on a conic the normals at which meet in a point, prove that the sum of the squares of the diameters parallel to AB and CD is equal to the sum of the squares of the diameters parallel to ^C'and BD. [Clare, 1885. 45. A parabola passes through two fixed points A, B Sit a distance 2a apart, and has a straight line distant c from the middle point of AB as directrix. Shew that the locus of the focus of the parabola is a conic section, which is an ellipse or a hyperbola, according as c is greater or less than a. [Trix. 1884. 46. A circle is drawn on a sheet of paper and the paper is folded so that one corner of the sheet lies on the circumference of the circle. Prove that as this corner moves about on the circle the crease on the paper will en- velope a conic. [Trix. 1884. 47. A semicircular piece of paper is folded over so that a particular point P on the bounding diameter lies on the circular boundary ; prove that the crease-line touches a fixed conic. [Trin. 1885. 48. If a circle and a conic intersect in the points B, C, D, E then the lines bisecting the angles between BG and DE, BD and CE, BE and CD are each parallel to one of two given straight lines. [Caius, 1885. 49. TP, TP' arc tangents to a conic, PG, P'G' are normals at P, P' : prove that TP : TP' :: PG : FG'.^ Provr also that if GL, G'L' are drawn perpendicular to PP', then PL= P'L'. [Chh. 1885. PROBLEMS. 199 50. Two tangents to a conic are drawn from any point T touching the conic in P and Q, any straight line drawn parallel to TP meets TQ in L, Pi^ in M and the conic in R, S : shew that W = Lli . LS. [Qu. 1885. 51. P, Q are any two points on an ellipse whose foci are S, H ; SP, HQ intersect in M, SQ, HP in N, and the bisectors of the angles QSP, QHP in li. Shew that RP, RQ are tangents to the ellipse, and M, N are points on a confocal hyperbola to which RM, RX are tangents. [Jes. 1885. 52. Given a line, a circle with centre 0, and a point *S*: a variable point R on the line is joined to /Sf by a line which meets the circle in U, V, and lines are drawn from >S' parallel to OU, OV to meet RO in points P and Q ; shew that the locus of these points is a conic with S as focus and the given line as directrix. Deduce from this mode of generation that tangents from any point to a conic subtend equal angles at a focus. [JOH. 1884. 53. Prove that the diagonals of a curvilinear quadri- lateral formed by the intersection of two confocal ellipses with two confocal hyperbolas are equal. Shew that these results are also true for a system of confocal and coaxial parabolas. [JoH. 1884. 54. A hyperbola is described having a focus of an ellipse for focus, and the tangent at the corresponding vertex for directrix. Prove that tangents to the ellipse from points in which the hyperbola cuts the minor axis of the ellipse are parallel to the asymptotes of the hvperbola. [JoH. 1884. 55. An ellipse and a hyperbola have the same foci and meet in P. PYZ is a tangent to the hyperbola at P; SY . HZ the focal perpendiculars. Prove that PY.PZ=BC\ where BOB' is the minor axis of the ellipse. [Pet. 1884. 56. An ellipse is met in P and Q by a rectangular hyperbola having for asymptotes the axes of the ellipse. 200 PROBLEMS. PM, QN are ordinates drawn to the axis CA ; PR, QT to CB. Prove that CM'-^CN'= CA\ and that CF : CR :: CA : CB. [Pet. 1884. 57. From a fixed point on the circumference of a circle a chord OA is drawn, and produced to B so that the difference of the squares on OB and OA is constant, prove that the line through B perpendicular to OB will touch a conic of which is centre and the other extremity of the diameter of the circle through is a focus. [Clare, 1884. 58. Given a focus S and two tangents to a conic, prove that the envelope of the minor axis is a parabola of which the focus is S. [Trin. 1884. 59. A focal chord PSQ of a conic is given in position and the position of the axis is also given. Trace the conic. [Pemb. 1884. 60. Prove by projection that, if AC A' be the major axis of an ellipse, and PNP' a double ordinate bisecting CA' at Ny the tangent at P is parallel to AP'. [Pemb. 1884. 61. An ellipse and a hyperbola are concentric and co- axial, and a point P is such that its polars with respect to the two are at right angles and intersect in Q ; prove that the locus of P is two straight lines through the centre C, and the locus of Q is two other straight lines through the centre ; but that if the conies be confocal, C, Q and P are in one straight line and CP . CQ is constant. [Chr. 1884. 62. Given the focus, directrix and eccentricity, give a geometrical construction for the points where a given straight line drawn through the focus cuts the curve. [Qr. 1884. 63. PQ is any chord of a conic, PG, QH the normals, G, H being on the axis, GL, HK are perpendiculars on PQ, shew that PL = QK. [Cath. 1884. 64. Prove that li A, B, G are three given points, two parabolas can be drawn thiougli ^l and B with C as focus, and that the axes of these parabolas are parallel to the asymptotes of the hyperbola which can be drawn through C with its foci at A and B. [Trin. 1885. PllOBLEMS. 201 65. If a parabola, having its focus coincident with one of the foci of an ellipse, touches the conjugate axis of the ellipse, a common tangent to the ellipse and parabola will subtend a right angle at the focus. [Trin. 1885. 06. AC A' and BCB' are the transverse and conjugate axes of an ellipse, of which S and S' are the foci, P is one of the points of intersection of this ellipse and a confocal hyper- bola, and aCa is the transverse axis of the hyperbola. Prove that SP = Aa, S'P=A'a, and aB = CP. [Trix. 1885. 07. Two fixed points P, Q are taken in the plane of a given circle, and a chord RS of the circle is drawn parallel to PQ, prove that for different positions of PS the locus of the point of intersection of EP and SQ is a conic. [Trix. 1880. 08. A circle passes through a fixed j^oint and cuts a i^iven straisfht line at a constant ano^le. Prove that the locus of the centre is a conic. [Jes. 1884. 09. A chord of a conic subtends a criven ano^le at the focus. Prove that the tanorents at its extremities will inter- sect on a conic having the same focus and directrix as the original conic. [JoH. 1883. 70. An ellipse and hyperbola have the same transverse axis, and their eccentricities ai'e the reciprocals of one an- other; prove that the tangents to each through the focus of the other intersect at right angles in two points and also meet the conjugate axes on the auxiliary circle. [JoH. 1884. 71. From any point Q on a central conic, QS, QH are drawn to the foci S, H, meeting the conic again in P, P' ; shew that if the tangents at P, P' meet in T, QT is bisected by the minor axis and the locus of T is a conic. [Pet. 1883. 72. Through two points on a central conic shew that two circles can be described to touch the conic; and that the points of contact are at the extremities of a diameter. [Caius, 1883. 202 PROBLEMS. CONE. 1. If >Sf be a point within the cone ; A its vertex, AB its axis; shew that the difference of the acute angles made with AB by the planes of the sections having S for a focus is twice the angle SAB. [I. C. S. 1887. 2. Shew how to obtain from a given cone a section which shall have the greatest possible eccentricity. [I. C. S. 188G. 3. Under what circumstances may the section of a cone by a plane be a rectangular hyperbola ? In such a case shew how to determine the necessary inclination of the cutting plane. [I. C. S. 1885. 4. Shew how to find the centre and the asymptotes of a hyperbolic section of a cone. Also shew how to cut from a given cone a hyperbola, whose asymptotes shall contain the greatest possible angle. [I. C. S. 1884. 5. Prove that the minor axis of an elliptic section of a right cone is a mean proportional between the diameters of the circular sections of the cone, made by planes drawn through the extremities of the major axis of the ellipse. If the ellipse be projected upon a plane perpendicular to the axis of the cone, shew that the distance between the foci of the curve of projection is equal to the difference between the radii of the same two circular sections. 6. From a given right circular cone is cut a series of parabolas the axes of which intersect a given straight line OM which passes through the vertex 0. If any section intersect OM at N, shew that the ratio ON'' : AN.CL is constant for all the parabolas, where A is the vertex of the section and (7 the centre of its focal sphere, and L is the point where the section cuts the axis OL of the cone. [Pemb. 1887. 7. If two sections of a cone have a common directrix, the latera recta of the sections are in the ratio of their eccen- tricities. [Jes. &c. 1888. 8. Prove that the locus of the centres of all plane sections, for which the distance between the foci is the same, is a right circular cylinder. [JoH. 1888. PROBLEMS. 203 9. Prove that the centres of all sections having their minor axis of the same length lie on the surface f e.XA or SA; .'. SK>SN. Case 2. If N is between S and A'. N X A S A' SK=e.XN, and SA' = e.XA'; :. by subtraction KA' = e . NA' < NA ' ; .-. SK>SN, Case 3. If N is in SA' produced. K N X A S A' SK=e.XN, and SA' = e.XA'; .: by subtraction A'K=e .A'N<:A'N; :. SKXA', .-. SKA'N, .'. SK>SN. Case 4. If A'' is between A and S. K A' X A S SK=e.NX>e.AX or SA; .-. sK>sy. Case 5. If A^ is in AS produced. .V A' X A SK = e . XN > XN > SN. We have now proved the circle does not intersect the perpendicular XP, when N is in any part the axis A A' between A and A', but they do intersect when N lies outside the part AA', hence the hyperbola lies entirely outside the lines drawn through A and A' at right angles to the axis. Cambritirjr : PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN ThIs BOOK ON THE DATE ^ ^ -_\" ^^^^^^^^ WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. MAR 19 1933 rc-n Lo MAR 1 ^m r,.AV 2 3 VsoQ ..^^^^^''' |P5iM''62^*^ '^ H0T6^nIW f^ECU LO APR 2 4 mi ^^^ J e 1%2 REC'D L[) SAug**^ APR 24 195^ REC'D LO |\PR Z4 1959 24Way60WW LD 21-50m-l,'38