CORNeu UNNBmrt UBRARIES Mathematict Library White Hall CORNELL UNIVERSITY LIBRARY 3 1924 059 322 481 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059322481 Production Note Cornell University Library pro- duced this volume to replace the irreparably deteriorated original. It was scanned using Xerox soft- ware and equipment at 600 dots per inch resolution and com- pressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Stand- ard Z39. 48-1984. The production of this volume was supported in part by the Commission on Pres- ervation and Access and the Xerox Corporation. 1991. fyxntW Winivtmi^ Jilrtatg THE GIFT OF ty\..V^ »^ MAIHEaiATICS tS Jt /f3 George Bell & Sons' CAMBRIDGE MATHEMATICAL SERIES. Crown 8vo. ARITHMETIC. With 8000 Examples. 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"FN the present Treatise the Conic Sections are defined -■- with reference to a focus and directrix, and I have en- deavoured to place before the student the most important properties of those curves, deduced, as closely as possible, from the definition. The construction which is given in the first Chapter for the determination of points in a conic section possesses several advantages; in particular, it leads at once to the constancy of the ratio of the square on the ordinate to the rectangle under its distances from the vertices; and, a|pain, in the case of the hyperbola, the directions of the asymptotes follow immediately from the construction. In several cases the methods employed are the same as those of Wallace, in the Treatise on Conic Sections, published in the Encyclopaedia Metropolitana. The deduction of the properties of these curves from their definition as the sections of a cone, seems d priori to be the natural method of dealing with the subject, but experience appears to have shewn that the discussion of conies as defined by their plane properties is the most suitable method of commencing an elementary treatise, and IV PREFACK TO THE FIRST EDITION. accordingly I follow the fashion of the time in taking that order for the treatment of the subject. In Hamilton's book on Conic Sectimis, published in the middle of the last century, the properties of the cone are first considered, and the advantage of this method of commencing the subject, if the use of solid figures be not objected to, is especially shewn in the very general theorem of Art. (156). I have made much use of this treatise, and, in fact, it contains most of the theorems and problems which are now re- garded as classical propositions in the theory of Conic Sections. I have considered first, in Chapter I., a few simple properties of conies, and have then proceeded to the par- ticular properties of each curve, commencing with the para- bola as, ia some respects, the simplest form of a conic section. It is then shewn, in Chapter VI., that the sections of a cone by a plane produce the several curves in question, and lead at once to their definition as loci, and to several of their most important properties. A chapter is devoted to the method of orthogonal pro- jection, and another to the harmonic properties of curves, and to the relations of poles and polars, including the theory of reciprocal polars for the particular case in which the circle is employed as the auxiliary curve. For the more general methods of projections, of reci- procation, and of anharmonic properties, the student will consult the treatises of Chasles, Poncelet, Salmon, Townsend, Ferrers, Whitworth, and others, who have recently deve- PREFACE TO THE FIRST EDITION. V loped, with so much fulness, the methods of modem Geo- metry. I have to express my thanks to Mr K B. Worthington, of St John's College, and of the Indian Civil Service, for valuable assistance in the constructions of Chapter XI., and also to Mr E. HilJ, Fellow of St John's College, for his kindness in looking over the latter half of the proof-sheets. I venture to hope that the methods adopted in this treatise will give a clear view of the properties of Conic Sections, and that the numerous Examples appended to the various Chapters will be useful as an exercise to the student for the further extension of his conceptions of these curves. W. H. BESANT. Caubbisqe, March, 1869. PREFACE TO THE NINTH EDITION. IN the preparation of this edition I have made many alterations and many additions. In particular, I have placed the articles on Eeciprocal Folars in a separate chapter, with considerable expansions. I have also in- serted a new chapter, on Conical Projections, dealing how- ever only with real projections. The first nine chapters, with the first set of miscel- laneous problems, now constitute the elementary portions of the subject. The subsequent chapters may be regarded as belonging to higher regions of thought I venture to hope that this re-arrangement will make it easier for the beginner to master the elements of the subject, and to obtain clear views of the methods of geometry as applied to the conic sections. A new edition, the fourth, of the book of solutions of the examples and problems has been prepared, and is being issued with this new edition of the treatise, with which it is in exact accordance. W. H. BESANT. December X4, 1894. CONTENTS. PAGE iNTRODnCnON .... 1 CHAPTER I. The Construction of a Conic Section, and Oeneral . Proferties 3 CHAPTER II. The Parabola . ... 20 CHAPTER III. The Elufbe ... 50 CHAPTER IV. The Htferbola .87 CHAPTER V. The Rectanoclar Htferbola 126 CHAPTER VI. The Cylinder and the Cone ... 137 CHAPTER VII. The Sdiilariiy of Conicb, the Areas of Conicb, and Curvature 164 vm CONTEHTS. CHAPTER VIII. PAGB Orthooonal Projegtioii . ... .168 CHAPTER IX. Op Conics in General . ... .177 CHAPTER X. Ellipses as Roulettes and Glibsettes 184 MiSCELLANEODS PrOBLEUB. I. . . 192 CHAPTER XI. Harhonic Properties, Poles and Polarb . 201 CHAPTER XII. Reciprocal Polars . . 220 CHAPTER XIII. The Construction of a Conic from Given Conditions 234 CHAPTER XIV. The Oblique Ctlinder, the Oblique Cone, and the Conoids 250 CHAPTER XV. Comical Projection ... 264 Miscellaneous Problems. II 278 CONIC SECTIONS. INTRODUCTION. DEFINITION. If a straight line and a point be given in position in u plane, and if a point move in a plane in such a manner that its distance from the given point always bears the same ratio to its distance from the given line, the curve traced out by the moving point is called a Conic Section. The fixed point is called the Focus, and the fixed line the Directrix of the conic section. When the ratio is one of equality, the curve is called a Parabola. "When the ratio is one of less inequality, the curve is called an Ellipse. When the ratio is one of greater inequality, the curve is called an Hyperbola. These curves are called Conic Sections, because they can all be obtained from the intersections of a Cone by planes in different directions, a fact which will be proved hereafter. It may be mentioned that a circle is a particular case of an ellipse, that two straight lines constitute a particular case of an hyperbola, and that a parabola may be looked upon as the limiting form of an ellipse or an hyperbola, under certain conditions of variation in the lines and magnitudes upon which those curves depend for their form. B. c. s. 1 2 INTRODUCTION. The object of the following pages is to discuss the general forms and characters of these curves, and to determine their most important properties by help of the methods and relations developed in the first six books, and in the eleventh book of Euclid, and it will be found that, for this purpose, a knowledge of Euclid's Geometry is all that is necessary. The series of demonstrations will shew the characters and properties which the curves possess in common, and also the special characteristics wherein they differ from each other; and the continuity with which the curves pass into each other will appear from the definition of a conic section as a Locus, or curve traced out by a moving point, as well as from the fact that they are deducible from the intersections of a cone by a succession of planes. CHAPTER I. PROPOSITION I. The Construction of a Conic Section. 1. Take S as the focus, and from S draw 8X at right angles to the directrix, and intersecting it in the point X. Definition. This line 8X, produced both ways, is called the Axis of the Conic Section. In SX take a point A such that the ratio of SA to AX is equal to the given ratio ; then Ais & point in the curve. Def. The point A is called the Vertex of the curve. In the directrix EX take any point E, join EA, and ES, produce these lines, and through S draw the straight line SQ making with ES produced the same angle which ES produced makes with the axis SN. 1—2 CONIC SECTIONS. Let P be the point of iutersection of 8Q and EA pro- duced, and through P draw LPK parallel to NX, and inter- secting ES produced in L, and the directrix in K. Then the angle PLS is equal to the angle LSN and therefore to PSL ; Hence SP = PL. Also PL : AS :: EP : EA :: PK : AX; .-.PL -.PK :: AS : AX; and .-. SP -.PK-.-.AS: AX. The point P is therefore a point in the curve required, and by taking for E successive positions along the directrix we shall, by this construction, obtain a succession of points in the curve. If E be taken on the upper side of the axis at the same distance from X, it is easy to see that a point P will be obtained below the axis, which will be similarly situated with regard to the focus and directrix. Hence it follows that the axis divides the curve into two similar and equal portions. CONIC SECTIONS. Another point of the curve, lying in the straight line KP, can he found in the following manner. Through iS draw the straight line FS making the angle FSK equal to KSP, and let FS produced meet KP produced in P'. Then, since KS bisects the angle PSF, SP' : SP :: P'K : PK; :. SF : P'K :: SP : PK, and P' is a point in the curve. 2. Def. The Eccentricity. The constant ratio of the distance from the focus of any point in a conic section to its distance from the directrix is called the eccentricity of the conic section. The Lotus Rectum. If E be so taken that EX is equal to SX, the angle PSN, which is double the angle LSN, and therefore double the angle ESX, is a right angle. For, since EX = SX, the angle ESX = SEX, and, the angle SXE being a right angle, the sum of the two angles SEX, ESX, which is equal to twice ESX, is also equal to a right angle. Calling R the position of P in this case, produce RS to R', so that R'S=RS; then R' is also a point in the curve. Def. The straight line RSR' drawn through the focus at right angles to the ojms, and intersecting the curve in R and R', is called the Latus Rectum, It is hence evident that the form of a conic section is determined by its eccentricity, and that its magnitude is CONICS. determined by the magnitude of the Latus Rectum, which is given by the relation 8R : SX :: SA : AX. 3. Def. The straight line PN (Fig. Art. 1), drawn from any point P of the curve at right angles to the axis, and intersecting the axis in N, is called the Ordinate of the point P. If the line PN be produced to P' so that NP' = NP, the line PNP" is a double ordinate of the curve. The latus rectum is therefore the double ordinate passing through the focus. Def. The distance AN of the foot of the ordinate from the vei-tex is called the Abscissa of the point P. Def. The distance SP is coiled the focal distance of the point P. It is also described as the radius vector drawn from the focus. 4. We have now given a general method of constructing a conic section, and we have explained the nomenclature which is usually employed. We proceed to demonstrate a few of the properties which are common to all the conic sections. For the future the word conic will be employed as an abbreviation for conic section. Prop. II. If the straight line joining two points P, P' of a conic meet the directrix in F, the straight line FS will bisect the angle between PS and P'S produced. CONICS. No straight line can meet a conic in more Draw the perpendiculars PK, P'K' on the directrix. Then SP : SP' ■.: PK : P'K' :: PF : FF. Therefore F8 bisects the outer angle, at S, of the triangle PSP'. (Euclid VL, A.) Cor. If SQ bisect the angle PSP', it follows that FSQ is a right angle. 5. Prop. III. than two points. Employing the figure of Art. 4, let P be a point of the curve, and draw any straight line FP. Join SF, draw SQ at right angles to SF, and SP' making the angle QSP" equal to QSP; then P' is a point of the curve. For, since SF bisects the outer angle at S, SP' : SP :: P'F : PF, :: P'K' : PK or SP' : P'K' :: SP : PK, and therefore, P' is a point of the curve, also, there is no other point of the curve in the straight line FPP'. For suppose if possible P" to be another point ; then, as in Article (4), SQ bisects the angle P^fP" ; but SQ bisects the angle PSP" ; therefore P" and P are coincident. 6. Prop. IV. If QSQ' be a focal chord of a conic, and P any point of the conic, and if QP, Q'P Tneet the directrix in E and F, the angle ESF is a right angle. For, by Prop. II., SE bisects the angle P/SQ', and SF bisects the angle PiSQ; hence it follows that ESF is a right angle. This theorem will be subse- quently utilised in the case in which the focal chord Q'SQ is coincident with the axis of the conic. 8 FOCAL CHOBDS. 7. Prop. V. The straight lines joining the extremities of two focal chords intersect in the directrix. li PSp, FSp' he the two chords, the point in which PP" meets the directrix is obtained by bisecting the angle PSP' and drawing SF at right angles to the bisecting line SQ. But this line also bisects the angle pSp' ; therefore pp' also passes through F. The line SF bisects the angle PSp', and similarly, if QS pro- duced, bisecting the angle ^i^', meet the directrix in F, the two lines Pp', P'p will meet in F. It is obvious that the angle FSF' is a right angle. 8. Prop. VL The semi-latus rectum is the harmonic mean between the two segments of any focal chord of a conic. Let PSP" be a focal chord, and draw the ordinates PN, ^ FN'. Then, the triangles SPN, SP'N' being similar, X SP : SF :: SN : SN' ^' ■.:NX-SX:SX-2f'X ■.-.SP-SR-.SR-SF, ^ since SP, SR, SF are proportional to NX, SX, and NX. /J ^/ p / IV r 1 TANGENT. Cor. SmceSP-.SP- 8R -.-.SP .SF -.SP .SF -SR.SF, and SF:8R-SF::SP.SF :SR.SP-SP.SF, it follows that 8P.SF-SR. SF = SR .SP-SP. SF; SR.PF = 2SP.SF. Hence, if PSF, QSQ' are two focal chords, PF:QQ'::SP.SF:SQ.SQ'. 9. Prop. VII. A focal chord is divided harmonically at the focus and the point where it meets the directrix. Let PSF produced meet the directrix in F, and draw PK, P'K' perpendicular to the directrix, fig. Art. 8. Then PF -.FF-.-.PK: P'K' SP : SF PF-SF:SF-FF; that is, PF, SF, and FF are in harmonic progression, and the line PF is divided harmonically at S and F. 10. Definition of the Tangent to a curve. If a straight line, drawn through a point P of a curve, meet the curve again in P', and if the straight line be turned round the point P until the point F approaches indefinitely near to P, the uUimate position of the straight line is the tangent to the curve at P. 10 CONICS. Thus, if the straight line APP' turn round P until the points P and P* coincide, the line in its ultimate position PT is the tangent at P. Def. The normal at any point of a curve is the straight line drawn through the point at right angles to the tangent at that point. Thus, in the figure, PG is the normal at P. Pkop. VIII. ITie straight line, drawn from the focus to the point in which the tangent jneets the directrix, is at right angles to the straight line draumfrom the focus to the point of contact. It is proved in Art. (4) that, if FPP' is a chord, and if 8Q bisects the angle P8P', FSQ is a right angle. Let the point P' move along the curve towards P; then, as P" approaches to coincidence with P, the straight line FPP" approximates to, and ultimately becomes, the tangent TP at P. But when P' coincides with P, the line 8Q coincides with 8P, and the angle F8P, which is ultimately T8P, becomes a right angle. Or, in other words, the portion of the tangent, intercepted between the point of contact and the directrix, subtends a right angle at the focus. TANGENTS. 11 11. PaoP. IX. The tangent at the vertex is perpendicular to the axis. If a chord EAP be drawn through the vertex, and the point P be near the vertex, the angle PSA is small, and LSN, which is half the angle PSN^, is nearly a right angle. Hence it follows that when P approaches to coincidence with A, the point E moves off to an infinite distance and the line EAP, which is ultimately the tangent at A, becomes parallel to LSE, and is therefore per- pendicular to AX. 12. Prop. X. The tangents at the ends of a focal chord intersect on the directrix. For the line SF, perpendicular to SP, meets the directrix in the same point as the tangent at P ; and, since SF is also at right angles to SP', the tangent at P' meets the directrix in the same point F. Conversely, if from any point F in the directrix tangents be drawn, the chord of contact, that is, the straight line joining the points of contact, will pass through the focus and will be at right angles to SF. Cor. Hence it follows that the tangents at the ends of the latus rectum pass through the foot of the directrix. 12 CONICS. 13. Prop. XI. If a chord FP meet the directrix in F, and if the line bisecting the PSP' meet the curve in q and q, Fq and Fq will he the tangents at q and q'. Taking the figure of Art. 7, the line SQ meets the curve in q and q', and, since SF is at right angles to SQ, it follows, from Art. 12, that Fq and Fq' are tangents. Hence if from a point F in the directrix tangents he drawn, and also any straight line FPP' cutting the curve in P and P', the chord of contact will bisect the angle PSP'. 14. Prop. XII. If the tangent at any point P of a conic intersect the directrix in F, and the lotus rectum produced in D, SD:SF::SA -.AX. Join SK ; then, observing that FSP and FKP are right angles, a circle can be described about FSPK, and therefore the angles SFD, SKP are equal. Also the angle FSD = complement of DSP = SPK; .: the triangles FSD, SPK are similar, and SD : SF :: SP : PK ■.-.SA: AX. Cor. (1). If the tangent at the other end F of the focal chord meet the directrix in U, sd' : sf :: sa : ax ; .: sd=sd: Cor. (2). K DE be the perpendicular from D upon SP, the triangles SDE, SFX are similar, and SE : SX :: SD : SF SA -.AX SR-.SX; SE is equal to SR, the semi-latus rectum. TAUGENTS. 13 15. Prop. XIII. The tangents drawn from any point to a conic subtend equal angles at thefoctts. Let the tangents FTP, FTP' at P and F meet the directrix in F and F and the latus rectum in D and D'. Join ST and produce it to meet the directrix in K ; then KF'.SD Hence KT -.ST KF' : 8D'. SB : SB' SF : SF' by Prop. XII. the angles TSF, TSF are equal. KF : KF' But the angles FSP", F'SP are equal, for each is the complement oiFSF'; .-. the angles TSP, TSF are equal. CoE. Hence it follows that if perpendiculars TM, TM be let fall upon SP and SF, they are equal in length. For the two triangles TSM, TSM have the angles TMS, TSM respectively equal to the angles TM'S, TSM', and the side TS common ; and therefore the other sides are equal, and TM = TM'. 16. Prop. XIV. If from any point T in the tangent ai a point P of a conic, TM he drawn perpendicular to the focal distance SP, and TN perpendicidar to the directrix, SM -.TN-.-.SA lAX. 14 CONICS. For, if PK be perpendicular to the directrix and 8F be joined, SM:SP SM : TN ■.TF:FP -.TN-.PK; ■.SP:PK : 8A : AX. This theorem, which is due to Professor Adams, may be employed to prove Prop. XIII. For if, in the figure of Ai-t. (15), TM, TM' be the perpendiculars from T on SP and 8P, and if TN be the perpendicular on the directrix, SM and SM have each the same ratio to TN, and are therefore equal to one another. Hence the triangles TSM, TSM' are equal in all respects, and the angle PSP' is bisected by ST. 17. Prop. XV. To draw tangents from any point to a conic. Let T be the point, and let a circle be described about S as centre, the radius of which bears to TN the ratio of SA : AX ; then, if tangents TM, TM be drawn to the circle, the straight lines SM, SM', produced if necessary, will intersect the conic in the points of contact of the tangents from T. 18. Prop. XVI. If PG, the normal at P, meet the axis of the conic in G, SG : SP :: SA : AX. Let the tangent at P meet the directrix in F, and the latus rectum produced in D. NORMALS. 15 Then the angle SPG = the complement of SPF=PFS, and PSG = the complement of FSX = FSD ; :. the triangles SFD, SPG are similar, and 8G : SP :: SD : SF :: SA : AX, by Prop. XII. 19. Prop. XVII. If from G, the point in which the normal at P meets the aods, GL he drawn perpendicular to SP, the length PL is equal to the semi-latus rectum. Let the tangent at P meet the directrix in F, and join SF. Then PLG, PSF are similar triangles ; .: PL : LG .: SF : SP. ^ Also SLG and SFX are similar triangles ; .-. LG -.SX::, Hence PL : SX :: , but SR -.SX::, .:PL = , ^ :/ P^ ?(? SA SA SR. SF. SP : AX, Art. (18), : AX, Art. (2) ; 20. Prop. XVIII. If from any point F in the directrix tangents he drawn, and also any straight line FPP cutting the curve in P and P', the chord PP' is divided harmonically at F and its paint of intersection with the chord of contact. 16 CONICS. For, if QSQ' be the chord of contact, it bisects the angle P.S'P', (Prop. XI.), and .■., if F be the point of intersection of SQ and PP', FF -.FP:: HP ■ SP P'V:PV FP'-FV:FV-FP. Hence FV is the harmonic mean between FP and FP'. The theorems of this article and of Art. 9 are particular cases of more general theorems, which will appear hereafter. 21. Prop. XIX. If a tangent be drawn parallel to a chord of a conic, the jjortion of this tangent which is inter- cepted by the tangents at the ends of the chord is bisected at the point of contact. M j?^ Let PP' be the chord, TP, TP' the tangents, and EQE' the tangent parallel to PP'. From the focus S draw SP, SP' and SQ, and draw TM, TM' perpendicular respectively to SP, SP'. Also draw from E perpendiculars EN, EL, upon SP, SQ, and from E' perpendiculars E'N', E'L' upon SP' and SQ. TAJUGENTS. 17 Then, since EM is parallel to PF TP : EP :: TF : EF, but TP : EP :: TM : EN, and TF : E'F :: TM' : E'N' ; .: TM : EN :: TM' : E'N' ; but TM=TM', Cor. Prop. XIII.; :.EN=E'N'. Again, by the same corollary, EN = EL and E'N' = E'L' ; .■.EL=E'L', and, the triangles ELQ, E'L'Q being similar, EQ^E'Q. CoE. If TQ be produced to meet PF in V, PV : EQ :: TV : TQ, and FViE'Qi-.TV-.TQ; .: PV=FV, that is, PF is bisected in V. Hence, if tangents be drawn at the ends of any chord of a conic, the point of intersection of these tangents, the middle point of the chord, and the point of contact of the tangent parallel to the chord, all lie in one straight line. EXAMPLES. 1. Describe the relative positions of the focus and directrix, first, when the conic is a circle, and secondly, when it consists of two straight lines. 2. Having given two points of a conic, the directrix, and the eccentricity, determine the conic. 3. Having given a focus, the corresponding directrix, and a tangent, construct the conic. B^ r. S 2 18 EXAMPLES. 4. If a circle passes through a fixed point and cuts a given straight line at a constant angle the locus of its centre is a conic. 5. If PG, pg, the normals at the ends of a focal chord, intersect in O, the straight line through parallel to Pp bisects Gg. 6. Find the locus of the foci of all the conies of given eccentricity which pass through a. fixed point P, and have the normal PO given in magnitude and position. 7. Having given a point /* of a conic, the tangent at P, and the directrix, find the locus of the focus. 8. If PSQ be a focal chord, and X the foot of the directrix, XP and XQ are equally inclined to the axis. 9. If PK be the perpendicular from a point P of a conic on the directrix, and iSK meet the tangent at the vertex in E, the angles SPE, KPE are equal. 10. If the tangent at P meet the directrix in F and the axis in T, the angles KSP, FTS are equal 11. PSP" is a focal chord, FN, PN' are the ordinates, and PK, P'K' perpendiculars on the directrix ; if KN, K'N' meet in Z, the triangle LNN' is isosceles. 12. The focal distance of a point on a conic is equal to the length of the ordinate produced to meet the tangent at the end of the latus rectiuu. 13. The normal at any point bears to the aemi-latus rectum the ratio of the focal distance of the point to the distance of the focus from the tangent. 14. The chord of a conic is given in length ; prove that, if this length exceed the latus rectum, the distance firom the directrix of the middle point of the chord is least when the chord passes through the focus. 15. The portion of any tangent to a conic, intercepted between two fixed tangents, subtends a constant angle at the focus. 16. Given two points of a conic, and the directrix, find the locus of the focus. 17. From any fixed point in the axis a line is drawn perpendiculair to the tangent at P and meeting SP in R ; the locus of ii is a circle. 18. If the tangent at the end of the latus rectum meet the tangent at the vertex in T, AT=AS. 19. TP, TQ are the tangents at the points P, Q of a conic, and PQ meets the directrix in II ; prove that &ST is a right angle. 20. SR being the semi-latus rectum, if RA meet the directrix in E, and SE meet the tangent at the vertex in T, AT=AS. EXAMPLES. 19 21. If from any point T, in the tangent at P, TM be drawn perpendicular to SP, and TN perpendicular to the transverse axis, meeting the curve in R, SM=8R. 22. If the chords P§, PQ meet the directrix in F and P, the angle /"Sif'ishalfPiSP'. 23. If PN be the ordinate, PQ the normal, and GL the perpen- dicular from G upon SP, GL : PN :: SA : AX. 24. If normals he drawn at the ends of a focal chord, a line through their intersection parallel to the axis will bisect the chord. 25. If a conic of given eccentricity is drawn touching the straight line FD joining two fixed points P and D, and if the directrix always passes through P, and the corresponding latus rectum always passes through J), &d the locus of the focus. 26. If ST, making a constant angle with SP meet in T the tangent at P, prove that the locus of 7* is a conic having the same focus and directrix. 27. If ^ be the foot of the perpendicular let fall upon PSP from the point of intersection of the normals at P and P', PE= SP and PP= SP. 28. If a circle be described on the latus rectum as diameter, and if the common tangent to the conic and circle touch the conic in P and the circle in Q, the angle PSQ is bisected by the latus rectum. (Eefer to Cor. 2. Art. 14.) 29. Given two points, the focus, and the eccentricity, determine the position of the axis. 30. If a chord PQ subtend a constant angle at the focus, the locus of the intersection of the tangents at P and § is a conic with the same focus and directrix. 31. The tangent at a point P of a conic intersects the tangent at the fixed point P' in Q, and from SQ, hat SP+PQ = PK-tPQ = QK. :.SQ8P, .: 8Q' > Q'K. If Q' lie in PK produced, SQ' + 8P>Pq, and .\SQ'>KQ'. 24. Prop. II. The Lotus Recttim = 4 . AS. For if. Fig. Art. 23, LSL' be the Latus Rectum, drawing LK' at right angles to the directrix, we have LS'=LK' = SX = 2AS. .-.LSL' =^4,. AS. 25. Mechanical construction of the Parabola. Take a rigid bar EKL, of ^ which the portions EK, KL are at right angles to each other, .£- and fasten a string to the end L, the length of which is LK. Z . Then if the other end of the string be fastened to jSi, and the bar be made to slide along a fixed straight edge, EKX, a pencil at P, keeping the string stretched against the bar, will trace out a portion of a parabola, of which S is the focus, and EX the directrix. THE PABABOLA. 23 26. Prop. III. If PK is the perpendicular upon the directrix from a point P of a parahola, and if PA meet the directrix in E, the angle KSE is a right angle. Join ES, and let KP and E8 produced meet at L. J^V Since 8A=AX, it follows tha.tPL=PK=8P; :. P is the centre of the circle through K, 8, and L, and the angle K8L is a right angle. 27. Prop. IV. parabola. Therefore KSE is a right angle. If PN is the ordinate of a point P of a PN" = iAS . AN. Taking the figure above, PN : EX :: AN : AX .: PN' -.EX.KX:: 4>A8 . AN : 4AS'. But, since K8E is a right angle, EX .KX=8X'' = 4:A8', .\PN' = 4,A8 .AN. CoK. If AN increases, and becomes infinitely large, PN increases and becomes infinitely large, and therefore the two portions of the curve, above and below the axis, proceed to infinity. 28. Prop. V. If from the ends of a focal chord per- pendiculars be let fall upon the directrix, the intercepted portion of the directrix subtends a right angle at the focus. For, if PA meet the directrix in E, and if the straight line through E perpendicular to the directrix meet P8 in P', it is shewn, in Art. 22, that P" is the other extremity of the focal chord P8 ; and, as in Art. 26, K8E is a right angle. 21 THE PARABOLA. 29. Prop. VI. The tangent at any point P bisects the angle between the focal distance SP and the perpendicular PK on the directrix. Let F be the point in which the tangent meets the directrix, ^ and join SF. We have shewn, (Art. 10) that FSP is a right angle, and, since SP = PK, and PF is common to the right-angled triangles SPF, KPF, it follows that these triangles are equal in all respects, and there- fore the angle SPF^'FPK. In other words, the tangent at any point is equally inclined to the focal distance and the axis. Cor. It has been shewn, in Art. (12), that the tangents at the ends of a focal chord intersect in the directrix, and therefore, if PS produced meet the curve in P', FP' is the tangent at P", and bisects the angle between SP" and the perpendicular from P' on the directrix. 30. Prop. VII. The tangents at the ends of a focal chord intersect at right angles in the directrix. Let P8P' be the chord, and PF, PF the tangents meeting the directrix in F. Let fall the perpendiculars PK, P'K', Aud join SK, SK'. The angle FSK' = ^P'SX = ^SPK = SPF, :. SK" is parallel to PF, and, similarly, 8K is parallel to P'F. But (Art. 28) KSK' is a right angle ; .•, PFP' is a right angle. THE PARiBOLA. 25 31. Prop. VIII. If the tangent at any point P of a parabola meet the axis in T, and PN he the ordinate of P, then AT=AN. Draw PK perpendicular to the directrix. The angle 5P2'=rPir = PTS, .■.ST=8P = PK = NX. X A\ But ST=SA+AT. and NX=AN+AX; .: since SA = AX, AT = AN. Def. The line NT is called the sub-tangent. The sub-tangent is therefore twice the abscissa of the point of contact. 32. Prop. IX. The foot of the perpendicular from the focus on the tangent at any point P of a parabola lies on the tangent at the vertex, and the perpendicular is a mean pro- portional between SP and SA. Taking the figure of the previous article, join SK meeting PT in Y. Then SP = PK, and PF is common to the two triangles SPY.KPY; also the angle SPY= YPK; .: the a,ng\e3YP = PYK, and SY is perpendicular to PT. Also SY= KY, and SA = AX, :. A Y is parallel to KX. 26 THE PARABOLA. Hence, AY is at right angles to AS, and is therefore the tangent at the vertex. Again, the angle SPY=STY=SYA, and the triangles SPY, SYA are therefore similar ; :. SP : SY :: SY : SA, ovSY'=SP .SA. 33. Prop. X. In the parabola the subnormal is constant and equal to the semi-lotus Rectum. Def. The distance between the foot of the ordinate of P and the point in which the normal at P meets the cms is called the subnormal. K P Z ^ ^ \ T X A. S JH G In the figure PG is the normal and PT the tangent. It has been shewn that the angle SPK is bisected by PT, and hence it follows that SPL is bisected by Pff, and that the angle SPG = GPL = PGS ; hence SG = SP = ST = SA +AT=SA + AN = 2AS+SN; .'. the subnormal WG = 2AS. 34. Cor. If 01 be drawn perpendicular to SP, the angle G PI = the complement of SPT, =the complement of STP, =PGN, and the two right-angled triangles OPN, GPl have their angles equal and the side OP common ; hence the triangles are equal, and Pl^.NG^iAS = the semi-latus Bectum. It has been already shewn, (Art. 19), that this property is a general property of all oonics. THE PARABOLA. 27 35. Prop. XI. To draw tangents to a parabola from an external point. For this purpose we may employ the general construction given in Art. (17), or, for the special case of the parabola, the followiDg construction. K ^^^ ^ V T \v V / ^^ ^^^ } / T i I A. s N Let Q be the external point, join BQ,, and upon /SQ as diameter describe a circle intersecting the tangent at the vertex in Y and F'. Join PIQ, YQ, ; these are tangents to the parabola. Draw 8P, so as to make the angle YST? equal to YSA, and to meet YQ in P, and let fall the perpendicular TN upon the axis. Then, SYQ is a right angle, since it is the angle in a semicircle, and, T being the point in which (^Y produced meets the axis, the two triangles SYP, SYT are equal in all respects ; .-. ^P = -Sr, and YT=YP. But 4 7 is parallel to PJV; .•.AT=AN. Hence SP=8T=SA+AT =AX+AN = JVX, and P is a point in the parabola. Moreover, if PK be perpendicular to the directrix, the angle SPY=STP=YPK, and PY is the tangent at P. (Art. 29.) Similarly, by making the angle Y'SP' equal to ASY" we obtain the point of contact of the other tangent QY'. 28 THE PARABOLA. 36. Prop. XII. If from, a paint Q tangents QP, QP be drawn to a parabola, the two triangles SPQ, SQP", are similar, and SO is a mean proportional between SP and SP'. K Y ^ T X 7 ■ A. S A T Produce PQ, to meet the axis in T, and draw SY, SY perpendicularly on the tangents. Then Y and Y' are points in the tangent at A. The angle SPQ = BTY = SYA since S, Y, Y, Q are points on a circle, and SYA, SQP" are in the same segment. Also, hy the theorem of Art. (15), the angle P8Q = QSF; therefore the triangles P8Q, QSP' are similar, and SP : SQ :: SQ : Sr. 37. From the preceding theorem the following, which is often useful, immediately follows. If from any points in a given tangent of a parabola, tangents be drawn to the curve, the angles which these tangents make with the focal distances of the points from which they are dravm are all equal. For each of them by the theorem, is equal to the angle between the given tangent and the focal distance of the point of contact. Hence it follows that the locus of the intersection of a THE PARABOLA. 29 tangent to a parabola with a straight line dravm through the focus meeting it at a constant angle is a straight line. For if QPbe the moveable tangent, the angle SQP=SFQ, and therefore, if 8QP is constant, SP'Q is a given angle. The point P' is therefore fixed, and the locus of Q is the tangent P'Q. 38. Since the two triangles PSQ, QSP" are similar, we have PQ : P'Q :: SP : SQ and PQ : PQ :: SQ : SP, .-.PQ^ : P^ :: SP: SP"; that is, the squares of the tangents from any point are proportional to the focal distances of the points of contact. This will be found to be a particular case of a subsequent Theorem, given in Art. 51. 39. Prop. XIII. ITie external angle between two tangents is half the angle subtended at the focus by the chord of contact. Let the tangents at P and P' intersect each other in Q and the axis AUN in T and T. Join SP, SP'; then the angles 8PT, STP are equal, and .-. STP is half the angle PSN; similarly STT is half FSK But TQT' is equal to the difference between STP and ST'P', and is therefore equal to half the difference between PSN and rSJ^, that is to half the angle P8P'. Hence, joining SQ, TQT' is equal to each of the angles PSQ. FSQ. 30 THE PARABOLA. 40. Prop. XIV. The tangents drawn to a parabola from any point make the same angles, respectively, with the axis and the focal distance of the point. Let QP, QP' be the tangents; join SP. and draw QE parallel to the axis, and meeting SP in E. Then, if PQ meet the axis in T, the angle EQP'=STP = SPQ = SQP'. (Art. 37.) i.e. QP and QP" respectively make the same angles with the axis and with QS. 41. Conceive a parabola to be drawn passing through Q, having S for its focua, SN' for its axis, and its vertex on the same side of S aa the vertex A of the given parabola. Then the normal at Q to this new parabola bisects the angle SQE; therefore the angles which QP and QP make with the normal at Q are equaL Hence the theorem, If from any point in a parabola, tangents he drawn to a eonfocal and co-axial parabola, the normal at the point will bisect the angle between the tanffents. If we produce SP to any point p, and take St equal to Sp, pt will be the tangent at p to the confociaJ and co-axial parabola passing through p. Hence the theorem, If parallel tangents be drawn to a series of eonfoccd and co-axial parabolas, the points of contact will lie in a straight line passing through the focus. In these enunciations the words co-axial and eonfocal are intended to imply, not merely the coincidence of the axes, but also that the vertices of the two parabolas are on the same side of their common focus. The reason for this will appear when we shall have discussed the analogous property of the eUipae. THE PARABOLA. 31 42. If two confocal parabolas have their axes in the same straight line, and their vertices on opposite sides of the focus, they intersect at right angles. For the angle and TPS=\PST, T'PS=iPST, .: TPT'=i{PST+PST')=a. right angle. It will be noticed that, in this case, the common chord PQ is equidistant from the directrices. For the distance of P from each directrix is equal to SP. 43. Prop. XV. The circle passing through the points of intersection of three tangents passes also through the focus. Let Q, P, Q' be the three points of contact, and F, T, F' the intersections of the tangents. 32 THE PAKABOLA. In Art. (36) it has been shewn that, if FP, FQ be tan- gents, the angle SQF=SFF. Similarly TQ, TQ' being tangents, the angle SQT=STQ', hence the angle SFF' or SFF = SQT, = STF', and a circle can be drawn through S, F, T, and F'. 44. Def. a straight line drawn parallel to t/ie axis through any point of a parabola is called a diameter. Prop. XVI. If from any point T tangents TQ, TQ' he drawn to a parabola, the point T is equidistant from the diameters passing through Q and Q', and the diameter drawn through the point T bisects the chord of contact. Join SQ, 8Q', and draw TM, TM' perpendicular re- spectively to SQ and SQ'. Also draw NTW per- pendicular to the diameters through Q and Q', and meeting those diameters in JV and N'. Then, since TS bisects the angle QSQ', r TM=TM'; and, since TQ bisects the angle SQl^, TN=TM. Similarly TN' = TM', .■.TN=TN'. Again, join QQ', and draw the diameter TV meeting QQ' in F; also let QT produced meet Q'N' in R ; then QV : YQ :: QT : TR :: TN: TN', since the triangles QTN, RTN' are similar ; .-. QV= VQ'. 9^ THE PARABOLA. 33 Hence the diameter through the middle point of a chord passes, when produced, through the point of intersection of the tangents at the ends of the chord. It should be noticed that any straight line drawn through T and terminated by QJV and Q'N' is bisected at T. 45. Pkop. XVII. Any diameter bisects all chords parallel to the tangent at its extremity, and passes through the point of intersection of the tangents at the ends of any of these chords. Let QQ' be a chord parallel to the tangent at P, and through the point of intersection T of the tangents at Q and Of draw FTF' parallel to QQ' and terminated at F and J" by the diameters through Q and Q'. Let the tangent at P meet TQ, TQ' in E and E', and QF, QF' in G and G'. Then EG : TF EQ : TQ E'Q' : TQ E'G' : TF'. But TF= TF'. since (Art. 44) T is equidistant from QG and Q'G', :.EG = E'G'. Also, EP = EG, since E is equidistant from QG and PV, the diameter at P. .: EP = E'P and GP = PG', and .■.QV=VQ'. B. c. s. 3 34 THE PARABOLA. Agaiu, since T, P, V are each equidistant from the parallel straight lines QF, Q'F', it follows that TPV is a straight line, or that the diameter VP passes through T. We have shewn that OE, EP, PE', EG' are all equal, and we hence infer that EE = :^GG' = hQQ!, and consequently that TP = J TV, or that TP = PV. Hence it appears, thai the diameter through the point of intersection of a pair of tangents passes through the point of contact of the tangeni parallel to the chord of contact, and also through the middle point of the chord of contact; and that the portion of the diameter between the point of inters&stion of the tangents and the middle point of the chord of contact is bisected at the point of contact of the parallel tangent. We may observe that in proving that EE" is bisected at P, we have demonstrated a theorem already shewn (Art. 21) to be true for all conies. 46. When the point T is on the directrix, QTQ^ is a, right angle. If then Qq is the chord which is normal at Q, it is parallel to the tangent 7Vi ^nd is therefore bisected by the diameter ^{7' through ^. Since QUia bisected by TV, it follows that i.e. the length of a normal chord is four times the portion of the parallel tangent between the directrix and the point of contact. THE PARABOLA. 35 47. Def. The line QV, parallel to the tangent at P, and terminated by the diameter PV, is called an ordinate of that diameter, and QQ' is the double ordinate. The point P, the end of the diameter, is called the vertex of the diameter, and the distance PV is called the abscissa of the point Q. We have seen that tangents at the ends of any chord intersect in the diameter which bisects the chord, and that the distance of this point from the vertex is equal to the distance of the vertex from the middle point of the chord. Def. The chord through the focus parallel to the tangent at any point is called the parameter of the diameter passing through the point. Prop. XVIII. The parameter of any diameter is four times the focal distance of the vertex of that diameter. Let' P be the vertex, and QfSQ' the parameter, T the point of intersection of the tangents at Q and Q', and FPF" the tangent at P. Then, since FS and FS bisect respectively the angles PSQ, PSQ, FSF' is a right angle, and, P being the middle point of FF', 8P = PF = PF'. Hence QQ', which is double FF', is four times SP. 48. Prop. XIX. If QVQf be a dmihle ordinate of a diameter PV, QV is a mean proportional between PV and the parameter of P. Let FPF' be the tangent at P, and draw the parameter through S meeting PF in TJ. The angle SUT = FPU = SPF' (Art. 29), and, since the angles SFQ, 8PF are equal (Art 36), it follows that the angles 8FT, SPF' are equal ; 3—2 36 THE PARABOLA. .-. SUT=SFT, and 17 is a point in the circle passing through SFTF'. Hence, QF being twice PF, QV' = iPF'' = 4PU.PT; but PU = SP, for the angle SUP = FPU= SPF' = PSU; and PT=PV, .: QTT' = 4!SP . PV. 49. This relation may be pre- sented in a different form, which is sometimes useful. If from any point U in the tan- gent at P, UQ IS drawn parallel to the axis, UP and VQ are respec- tively equal to the ordinate and abscissa of the point Q with I'egard to the diameter through P, and therefore PU'=iSP. C'Q. Therefore, if VR is drawn parallel to the axis from another point V of the tangent, PUi : PVi :: UQ : VR. THE PABABOLA. 37 Hence, since UE : VR:.PU: PV, XJEi : VB? :: Uq : VR :: UQ . VR : VR\ and UE^=UQ. VR. Hence UE : UQ :: VR : UE :: PR : PE ; .: UQ : QE :: PE : ER. In a similar manner it can be shewn that VF^= UQ . VR, and it follows that VF= UE, and therefore that EF is parallel to the tangent at P. •50. Prop. XX. If QVQ' be a double ordinate of a diameter PV, and QD the perpendicular from Q upon PV, QD is a mean proportional between PV and the latv^ rectum. Let the tangent at P meet the tangent at the vertex in Y, and join SY. The angle QVD=SPY=SYA, and therefore the triangles QVI), SA Y are similar ; and QD' : QV :: AS' : Sr :: AS' -.AS. SP -.-.AS :SP :: ^A8 . PV : ^SP . PV, but QV' = 4,SP.PV; :.QIf = ^AS.PV. 51. Prop. XXI. If from any point, within or without a parabola, two straight lines be drawn in given directions and intersecting the curve, the ratio of the rectangles of the segments is independent of the position of the point. From any point draw a straight line intersecting the 38 THE PARABOLA. parabola in Q and Q', and draw the diameter OE, meeting the curve in E. If PF be the diameter bisecting QQ', and EU the ordinate, OQ . OQ' = OV^ - QV = EU'-QT = 4>SP . PU -4SP . PV = 4>SP.0E. Similarly, if ORB! be any other intersecting line and P the vertex of the diameter bisecting RK, 0R.0R = 4>SF .OE. .: OQ.OQf :0R. OR :: SP : SP', that is, the ratio of the rectangles depends only on the positions of P and P', and, if the lines OQQf, ORR' are drawn parallel to given straight lines, these points P, P' are fixed. It will be easily seen that the proof is the same if the point be within the parabola. If the lines 0Q(^, ORR' be moved parallel to themselves until they become the tangents at P and P', we shall then obtain, if these tangents intersect in T, yps . yp'2 .. 8p . SP, a result previously obtained (Art. 38). Again if QSQ. RSR' be the focal chords parallel to TP and TP', it follows that .-. (cor. Art. 8) TP" : TP" :: QQ' : RR'. THE PARABOLA. 39 52. Prop. XXII. If from, a point 0, outside a para- bola, a tangent OM, and a chord OAB he drawn, and if the diameter ME meet the chord in E, OE^=OA .OB. Let P be the point of contact of the tangent parallel to OAB, and let OM, ME meet this tangent in T and F. Draw TV parallel to the axis and meeting PM in V; then OA .OB: OM" :: TP' : TM' (Art. 51), ::TF": TM", since PM is bisected in V; also TF:TM::OE:OM; .: OE"=OA .OB. Cor. 1. If AL, BN be the ordinates, parallel to OM, of A and B, ML, ME, and MN are proportional to OA, OE and OB, and therefore ME"=ML .MN. This theorem may be also stated in the following form : If a chord AB of a parabola intersect a diameter in the point E, the distance of the point E from the tangent at the end of the diameter is a mean proportional between the dis- tances of the points A and Bfrom the same tangent. 40 THE PABABOLA. Cor. 2. Let KE be the ordinate through E parallel to Then, since ML : ME :: ME : MN, AD : KE' :: KE^ : BN" .: AL : KE :: KE : BN, so that KE is a mean proportional between AL and BN, the ordinates of A and B. 53. Pkop. XXIII. If a circle intersect a parabola in four points, the two straight lines constituting any one of the three pairs of the chords of intersection are equally in- clined to the axis. Let Q, Q', R, R' be the four points of intersection ; then OQ . OQ = OR . OR', and therefore SP, SP' are equal, (Art. 51). But, if SP, SP" be equal, the points P, P' are on opposite sides of, and are equidistant from the axis, and the tangents at P and P' are therefore equally inclined to the axis. Hence the chords QQf, RR, which are parallel to these tangents, are equally inclined to the axis. In the same manner it may be shewn that QR, Q'R' are equally inclined to the axis, as also QR', Q'R. EXAMPLES. 41 54. Conversely, if two chords QQ^, RR', which are not parallel, make equal angles with the axis, a circle can be drawn through Q, §", R', R. For, if the chords intersect in 0, and OE be drawn parallel to the axis and meeting the curve in E, it may be shewn as above that OQ . 0^=4SP . OE, and OR . OR'=iSP' . OE, P and P being the vertices of the diameters bisecting the chords. But the tangents at P and P, which are parallel to the chords, are equally inclined to the axis, and therefore SP is equal to SP Hence OQ.O^=OR. OR', and therefore a circle can be drawn through the points Q, §", R, R'. If the two chords are both perpendicular to the axis, it is obvious that a circle can be drawn through their extremities, and this is the only case in which a circle can be drawn through the extremities of panillel chords. EXAMPLES. 1. Find the locus of the centre of a circle which passes through a given point and touches a given straight line. 2. Draw a tangent to a parabola, making a given angle with the axis. 3. If the tangent at P meet the tangent at the vertex in Y, AY^=AS.AN. 4. If the normal at P meet the axis in G, the focus is equidistant from the tangent at P and the straight line through G parallel to the tangent. 5. Given the focus, the position of the axis, and a tangent, construct the parabola. 6. Find the locus of the centre of a circle which touches a given straight line and a given circle. 7. Construct a parabola which has a given focus, and two given tangents. 8. The distance of any point on a parabola from the focus is equal to the length of the ordinate at that point produced to meet the tangent at the end of the latus rectum. 9. PT being the tangent at P, meeting the axis in T, and PN the ordinate, prove that TY . TP=TS. TN. 4Z EXAMPLES. 10. If SE be the perpendicular from the focus on the normal at P, ahew that 11. The locus of the vertices of all parabolas, which have a common focus and a common tangent, is a circle. 12. Having given the focus, the length of the latus rectum, and a tangent, construct the parabola. 13. If PSF be a focal chord, and PiV, P-J" the ordinates, shew that AiY,AN'=AS^. Shew also that the latus rectum is a mean proportional between the double ordinates. 14. The locus of the middle points of the focal chords of a parabola is another parabola. 15. Shew that in general two parabolas can be drawn having a given straight line for directrix, and passing through two given points on the same side of the line. 16. Pp is a chord perpendicular to the axis, and the perpendicular from p on the tangent at P meets the diameter through P in R; prove that ItP is equal to the latus rectum, and find the locus of R. 17. Having given the focus, describe a parabola passing through two given points. 18. The circle on any focal distance as diameter touches the tangent at the vertex. 19. The circle on any focal chord as diameter touches the directrix. 20. A point moves so that its shortest distance from a given circle is equal to its distance from a given diameter of the circle ; prove that the locus is a parabola, the focus of which coincides with the centre of the circle. 21. Find the locus of a point which moves so that its shortest distance from a given circle is equal to its distance from a given straight line. 22. The vertex of an isosceles triangle is fixed. The extremities of its base lie on two fixed parallel straight lines. Prove that the base is a tangent to a parabola. 23. Shew that the normal at any point of a parabola is equal to the ordinate through the middle point of the subnormal. 24. If perpendiculars are drawn to the tangents to a parabola where they meet the axis they will be normals to two equal parabolas. 25. PSP is a focal chord of a parabola. The diameters through P, P meet the normals at P, P m. V,V' respectively. Prove that PVV'P is a parallelogram. EXAMPLES. 43 26. If AFC be a sector of a circle, of which the radius CA is fixed, and a circle be described, touching the radii CA, CP, and the arc AP, the locus of the centre of this circle is a parabola. 27. If from the focus jS of a parabola, SY, SZ be perpendiculars drawn to the tangent and normal at any point, YZ is parallel to the diameter. 28. Prove that the locus of the foot of the perpendicular from the focus on the normal is a parabola. 29. If PG be the normal, and OL the perpendicular from O upon SP, prove that GL is equal to the ordinate PN. 30. Given the focus, a point P on the curve, and the length of the perpendicular firom the focus on the tangent at P, find the vertex. 31. A circle is described on the latus rectum as diameter, and a common tangent QP is drawn to it and the parabola : shew that SP, SQ make equal angles with the latus rectum. 32. G is the foot of the normal at a point P ai the parabola, Q is the middle point of SG, and X is the foot of the directrix: prove that QXi-Qpi=AAS\ 3.3. If PG the normal at P meet the axis in G, and if PF, PH, lines eqiially inclined to PG, meet the axis in F and H, the length SG is a mean proportional between SF and SH. 34. A triangle ABC circumscribes a parabola whose focus is iS', and through A, B,C, lines are drawn respectively perpendicular to SA, SB, SO ; shew that these pass through one point. 35. If PQ be the normal at P meeting the curve in Q, and if the chord PJi be drawn so that PR, PQ are equally inclined to the axis, PRQ is a right angle. 36. PiV is a semi-ordinate of a parabola, and AM in taken on the other side of the vertex along the axis equal to AN; from any point Q in PjV, QR is drawn parallel to the axis meeting the curve in R ; prove that the lines MR, AQ will intersect in the parabola. 37. Having given two points of a parabola, the direction of the axis, and the tangent at one of the points, construct the parabola. 38. Having given the vertex of a diameter, and a corresponding double ordinate, construct the parabola. 39. PM is an ordinate of a point P ; a straight line parallel to the axis bisects PM, and meets the cun-e in Q ; MQ meets the tangent at the vertex in T; prove that SAT=2PM. 40. AB, CD are two parallel straight lines given in position, and AC is perpendicular to both, A and C being given points; in CD any point Q is taken, and in AQ, produced if necessary, a point P is taken. 44 EXAMPLES. such that the distance of P from AB is equal to CQ ; prove that the locus of P is a parahola. 41. If the tangent and normal at a point P of a parabola meet the tangent at the vertes in K and L respectively, prove that KI^ : SP^ :: SP-AS : AS. 42. Having given the length of a focal chord, find its position. 43. If the ordinate of a point P bisects the subnormal of a point P', prove that the ordinate of P is equal to the normal of i*- 44. A parabola being traced on a plane, find its aiis and vertex. 45. If P F, i" T' be two diameters, and P F', P T ordinates to these diameters, PV=P'V'. 46. If one side of a triangle be parallel to the axis of a parabola, the other sides will be in the ratio of the tangents parallel to them. 47. QVQ^ is an ordinate of a diameter PV, and any chord PR meets Q^ in JV, and the diameter through Qin L; prove that PL^ = PJ\r.PR. 48. Describe a parabola passing through three given points, and having its axis parallel to a given line. 49. If .ilP, AQ be two chords drawn from the vertex at right angles to each other, and PN, QM be ordinates, the latus rectum is a mean proportional between AN and AM. 50. PSp is a focal chord of a parabola ; prove that AP, Ap meet the latus rectum in two points whose distances from the focus are equal to the ordinates oip and P respectively. 51. If the straight line AP and the diameter through P meet the double ordinate QMQ! in R and R', prove that RM.RM=QMK 52. A and P are two fixed points. Parabolas are drawn all having their vertices at A, and all passing through P. Prove that the points of intersection of the tangents at P with the tangent and normal at A lie on two fixed circles, one of which is double the size of the other. 53. A variable tangent to a parabola intersects two fixed tangents in the points T and T' : shew that the ratio ST . ST' is constant. 54 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. 55. If SY be the perpendicular on the tangent at P, and if YS be produced to A so that SR=SY, shew that PAR is a right angle. EXAMPLES. 45 56. If two circles be drawn touching a parabola at the ends of a focal chord, and passing through the focus, shew that they intersect each other orthogonally. 57. PSQ is a focal chord of a parabola, whose vertex is A and focus S, V being the middle point of the chord, shew that PVi=Ar^+3AJSP. 58. QQ^ is a focal chord of a parabola. Describe a circle which shall pass through Q, Q and touch the parabola. If P be the point of contact and the angle QPQ' a right angle, find the inclination of QP to the axis. 59. Through two fixed points E, F, on the axis of a parabola are drawn two chords PQ, PR meeting the curve in P, Q, R. If QR meet the axis in T, shew that the ratio TR : TQ is constant. 60. A chord PQ is normal to the parabola at P, and the angle PSQ is a right angle. Prove that SQ=2SP, and that the ordinate of P is equal to the latus rectum. Also, if T is the point of intersection of the tangents at P and Q, and if R is the middle point of TQ, prove that the angle TSR is a right angle, and that ST=2SR. 61. A straight line intersects a circle ; prove that all the chords of the circle which are bisected by the straight line are tangents to a parabola. 62. If two tangents TP, TQ be drawn to a parabola, the perpen- dicular S£ from the focus on their chord of contact passes through the middle point of their intercept on the tangent at the vertex. 63. From the vertex of a parabola a perpendicular is drawn on the tangent at any point ; prove that the locus of its intersection with the diameter through the point is a straight line. 64. If two tangents to a parabola be drawn from any point in its axis, and if any other tangent intersect these two in P and Q, prove that SP^SQ. 65. T' is a point on the tangent at P, such that the perpendicular from T on SP is of constant length ; prove that the locus of 3" is a parabola. If the constant length be 2 AS, prove that the vertex of the locus is on the directrix. 66. Given a chord of a parabola in magnitude and position, and the point in which the axis cuts the chord, the locus of the vertex is a circle. 67. If the normal at a point P of a parabola meet the curve in Q, and the tangents at P and Q intersect in T, prove that T and P are equidistant from the directrix. 46 EXAMPLES. 68. If TP, TQ be tangents to a parabola, such that the chord PQ is normal at P, PQ : PT :: JP.V : AN, PN and AN being the ordinate and abscissa. . 69. If two equal tangents to a parabola be cut by a thiid tangent, the alternate segments of the two tangents will be equal. 70. If AP be a chord through the vertex, and if PX^perpendicular to AP, and PG, the normal at P, meet the axis in L, (r respectively, G-'Z=half the latus rectum. 71. If PSQ be a focal chord, A the vertex, and PA, QA be produced to meet the directrix in i*, §' respectively, then P'SQ' will be a right angle. 72. The tangents at P and Q intersect in T, and the tangent at R intersects TP and TQ in C and D; prove that PC : CT :: CR : RD :: TB : DQ. 73. From any point D in the latus rectum of a parabola, a straight line DP is drawn, parallel to the axis, to meet the curve in P ; if X be the foot of the directrix, and A the vertex, prove that AD, XP intersect in the parabola. 74. PSp is a focal chord, and upon PS and pS as diameters circles are described ; prove that the length of either of their common tangents is a mean proportional between AS and Pp. 75. li AQ he & chord of a parabola through the vertex A, and QR be drawn perpendicular to AQ to meet the axis ia R; prove that AR will be equal to the chord through the focus parallel to AQ. 76. If from any point P of a circle, PC be drawn to the centre C, and a chord PQ be drawn parallel to the diameter AB, and bisected in R ; shew that the locus of the intersection of CP and AR is a parabola. 77. A circle, the diameter of which is three-fourths of the lattis rectum, is described about the vertex .i of a parabola as centre ; prove that the common chord bisects AS. 78. Shew that straight lines drawn perpendicular to the tan- gents of a parabola through the points where they meet a given fixed line perpendicular to the axis are in general tangents to a confocal parabola. 79. If QR be a double ordinate, and PD a straight line drawn parallel to the axis from ajiy point P of the cinr'e, and meeting QR in D, prove, from Art. 27, that QD . RD=4AS . PD. 80. Prove, by help of the preceding theorem, that, if QQ be a chord parallel to the tangent at P, QQ' is bisected by PD, and hence determine the locus of the middle point of a series of parallel chords. EXAMPLES. 47 81. If a parabola touch the sides of an equilateral triangle, the focal distance of any vertex of the triangle passes through the point of contact of the opposite side. 82. Find the locus of the foci of the parabolas which have a common vertex and a common tangent. 83. From the points where the normals to a parabola meet the axis, lines are drawn perpendicular to the normals : shew that these lines will be tangents to an equal parabola. 84. Inscribe in a given parabola a triangle having its sides parallel to three given straight lines. 85. PNI" is a double ordinate, and through a point of the parabola RQL is drawn perpendicular to PP' and meeting PA, or PA produced in R ; prove that PN : NL :: LR : RQ. 86. PNP is a double ordinate, and through R, a point in the tangent at P, RQM is drawn perpendicular to PP and meeting the curve in Q ; prove that QM : QR :: P'M : PM. 87. If from the point of contact of a tangent to a parabola, a chord be drawn, and a line parallel to the axis meeting the chord, the tangent, and the curve, shew that this line will be divided by them in the same ratio as it divides the chord. 88. PSp is a focal chord of a parabola, RD is the directrix meet- ing the axis in D, Q is any point in the curve ; prove that if QP, (^ produced meet the directrix in R, r, half the latus rectum will be a mean proportional between DR and Dr. 89. A chord of a parabola is drawn parallel to a given straight Une, and on this chord as diameter a circle is described ; prove that the distance between the middle points of this chord, and of the chord joining the other two points of intersection of the circle and parabola, will be of constant length. 90. If a circle and a parabola have a common tangent at P, and intersect in Q and R; and if QV, UR be drawn paraUel to the axis of the parabola meeting the circle in V and U respectively, then will FiTbe parallel to the tangent at P. 91. If PV be the diameter through any point P, QV a, semi- ordinate, §* another point in the curve, and QfP cut QV ia. R, and QfR', the diameter through §*, meet QVin R', then VR . VR'=QV^. 92. PQ, PR are any two chords ; PQ meets the diameter through R in the point F, and PR meets the diameter through Q in E; prove that EF is parallel to the tangent at P. 48 EXAMPLES. 93. If parallel chorda be intersected by a diameter, the distances of the points of intersection from the vertex of the diameter are in the ratio of the rectangles contained by the segments of the chords. 94. If tangents be drawn to a parabola from any point P in the latus rectum, and if §, §" be the points of contact, the semi-latas rectum is a geometric mean between the ordinates of Q and Q", and the distance of P from the axis is an arithmetic mean between the same ordinates. 95. If A', K, C be the middle points of the sides of a triangle A.BC, and a parabola drawn through A', B', C meet the sides again in A", B", C", then will the Unes AA", BB", CO" be parallel to each other. 96. A circle passing through the focus cuts the parabola in two points. Prove that the angle between the tangents to the circle at those points is four times the angle between the tangents to the parabola at the same points. 97. The loc\is of the points of intersection of normals at the extremities of focal chords of a parabola is another parabola. 98. Having given the vertex, a tangent, and its point of contact, construct the parabola. 99. PSp is a focal chord of a parabola ; shew that the distance of the point of intersection of the normals at P and p from the directrix varies as the rectangle contained by PS, pS. 100. TP, TQ are tangents to a parabola at P and Q, and is the centre of the circle circumscribing PTQ ; prove that TSO is a right angle. 101. P is any point of a parabola whose vertex is A, and through the focus S the chord QSQ' is drawn parallel to AP ; PN, QM, Q^M', being perpendicular to the- axis, shew that SM ia a mean proportional between AM, AN, and that MM'=AP. 102. If a circle cut a parabola in four points, two on one side of the axis, and two on the other, the sum of the ordinates of the first two is equal to the sum of the ordinates of the other two points. Extend this theorem to the case in which three of the points are on one side of the axis and one on the other. 103. The tangents at P and Q meet in T, and TL is the per- pendicular from T on the axis ; prove that if PN, QM be the ordinates of P and ft PN . QM=^S . AL. 104. The tangents at P and Q meet in T, and the lines TA, PA, QA, meet the directrix in t, p, and q : prove that tp=tq. EXAMPLES. 49 105. From a point T tangents TP, TQ are drawn to a parabola, and through T straight lines are drawn parallel to the normals at P and Q ; prove that one diagonal of the parallelogram so formed passes through the focus. 106. Through a given point within a parabola draw a chord which shall be divided in a given ratio at that point. 107. ABC is a portion of a parabola bounded by the axis AB and the semi-ordinate BG : find the point P in the semi-ordinate such that if PQ be drawn parallel to the aads to meet the parabola in Q, the sum of BP and PQ shall be the greatest possible. 108. The diameter through a point i* of a parabola meets the tangent at the vertex in Z; the normal at P and the focal distance of Z will intersect in a point at the same distance from the tangent at the vertex as P. 109. Given a tangent to a parabola and a point on the curve, shew that the foot of the ordinate of the point of contact of the tangent drawn to the diameter through the given point lies on a fixed straight line. 110. Find a point such that the tangents from it to a parabola and the Unes from the focus to the points of contact may form a parallelogram. 111. Two equal parabolas have a common focus ; and, from any point in the common tangent, another tangent is drawn to each ; prove that these tangents are equidistant from the common focus. 112. Two parabolas have a common axis and vertex, and their concavities turned in opposite directions ; the latus rectum of one is eight times that of the other ; prove that the portion of a tangent to the former, intercepted between the common tangent and axis, is bisected by the latter. B. C. S. CHAPTER III. The Ellipse. Def. An ellipse is the curve traced out by a point which moves in such a manner thai its distance from, a given point is in a constant ratio of less inequality to its distance from a given straight line. Tracing the Curve. 55. Let S be the focus, EX the directrix, and SX the perpendicular on EX from 8. Divide SX at the point A in the given ratio ; the point A is the vertex. From any point E in EX, draw EAP, E8L, and through 8 draw 8P making the angle P8L equal to L8N, and meeting EAP in P. Through P draw LPK perpendicular to the directrix and meeting ESL in L. THE ELLIPSE. 51 Then the angle PSL = LSN = SLP. .:SF = PL. Also PL : PK :: SA : AX. I Hence 8P : PK :: 8A : AX, and P is therefore a point in the curve. Again, in the axis XAN find a point A' such that SA' : A'X :: SA : AX; this point is evidently on the same side of the directrix as the point A, and is another vertex of the curve. Join EA' meeting PS produced in P', and draw P'L'K' perpendicular to the directrix and meeting ES in L'. Then FL' ■ P'K' :: SA' : A'X :: SA : AX, and the angle SL'P' = L'SA = L'SF ; .: FL' = SF. Hence F is also a point in the curve, and PSP' is a focal chord. By giving E a series of positions on the directrix we shall obtain a series of focal chords, and we can also, as in Art. (1), find other points of the curve lying in the lines KP, K'F, or in these lines produced. We can thus find any number of points in the curve. 4—2 52 THE ELLIPSE. 56. Def. The distance AA' is the major axis. The middle paint G of AA' is called the centre of the Ifthrotigh C the double ordinate BCR be drawn, BB' is called the minor axis. Any straight line drawn through the centre, and terminated by the curve, is called a diam,eter. The lines AG A', BGB' are called the principal diameters, or, briefly, the axes of the curve. The line AG A' is also sometimes called the transverse axis, and BGB' the conjugate axis. 57. Prop. I. IfFbe any point of an ellipse, and A A' the axis major, and if PA, A'P, when produced, meet the directrix in E and F, the distance EF subtends a right angle at the focus. By the theorem of Art. 4, E8 bisects the angle ASP", and FS bisects the angle ASP ; :. ESF is a right angle. It will be seen that, since ASA' is a focal chord, this is a particular case of the theorem of Art. 6. THE ELLIPSE. 53 58. Prop. II. If PJV" he the ordinate of any point P of an ellipse, AG A' the aads major, and BOB' the ax-is minor, PN' : AF. NA' :: BC : AC Join PA, A'P, and let these lines produced meet the directrix in E and F. Then PUT : AF and PF : A'F .-.PF" : AF.FA' EX : AX, FX : A'X; EX.FX : AX. A'X SX' : AX. A'X, since ESF is a right angle (Prop, i.); that is, PF^ is to AF. FA' in a constant ratio. Hence, taking PF coincident with BG, in which case AF=FA'=AG, BG" : AC :: 8X* : AX. A'X, and .-. PF" ': AF . FA' :: BC : AC. This may be also written PF* : AC-CF" :: BC : AC. Cor. If PM be the perpendicular from P on the axis minor, CM = PF,PM=GF, and GJiP : AC-PJIP :: BC : AG". Hence AG' : AG*-PM' :: BG' : GJHP, and .-. AG' : PM' .: BG' : BG'- CM', or PM' : BM.MB :: AC : BG'. 54 THE ELLIPSE. 59. If a point N' be taken on the axis major, between G and A', such that ON' = GN, the corresponding ordinate P'N' = FN, and therefore it follows that the curve is sym- metrical with regard to BCB', and that there is another focus, and another directrix, corresponding to the vertex A'. 60. By help of the theorem of Art. 57, we can give an independent proof of the existence of the other focus and directrix, corresponding to the vertex A'. In AA' produced take a point X' such that A'X' = AX, and in AA' take a point S' such that A'S" = AS. Through X' draw a straight line eX'f perpendicular to the axis, and let EP, FP produced meet this line in e and /. Join e-S', and ^'. Then eX' : EX :: AX' : AX A'X : A'X' FX :fX'; ■.eX' . fX' = EX.FX = SX^ = S'X" THE ELLIPSE. 55 Hence eS'/is a right angle. Through P draw KPk parallel to the axis, meeting eS' and fS' produced in L and I. Then PL : Pk :: S'A : AX' :: SA' : A'X, and PI : Pk :: S'A' : A'X' :: SA : AX, .: PL = Pl. Moreover, LS'l being a right angle, ST = Pl, .-. S'P : Pk :: S'A' : A'X', and the curve can he described by means of the focus S' and the directrix eX'. If SA be equal to AX, the point A', and therefore the points SP+8'P. Euclid I. 21, i.e. SQ + 8'Q>AA'. If Q' be within the ellipse, let SQ^, S'Q^ produced meet the curve and take a point P on the inter- cepted arc. Then Q' is within the triangle 8PS', and ■.SP + 8'P>SQ' + 8'Q', i.e. SQ' + S'Q'A 0" = S'K' = ST + TK' = S'T'' + Sr = 2Cr + 2CS' (Euclid, ii. 12 and 13); and T lies on a fixed circle, of which C is the centre. This circle is called the Director Circle of the Ellipse, and it will be seen that when the ellipse, by the elongation of SC from S is transformed into a parabola, the director circle merges into the directrix of the parabola. Cor. If XQ is the tangent to the director circle from the foot of the directrix, XQ'=CX''-GQ'= CX'- -CA'- GB' = CX^-SC.CX-SC.SX (Arts. 61 and 63), = CX.SX-SC.8X = SX\ .■.XQ = SX, and hence it follows that the directrix is the radical axis of the director circle and of a point circle at thefoaus. 92. Prop. XXXI. The rectangles contained by the seg)nents of any two chords which intersect each other are in the ratio of the squares of the parallel diameters. Through any point (3 in a chord OQQ' draw the diameter ORR', and let CD be parallel to QQ', and CP conjugate to CD, bisecting QQ' in V. Draw RU parallel to CD. THE ELLIPSE. 77 Then CD'-RU' : CU' :: CD" : OP'- (Art. 82), :: CD'-QV : GV\ But RU' : CU' :: OV : GT ; .: CD' : GTP :: CD^ + OV-QV : GV or CD- : CD^ + OV-QV' :: GU' : OF' :: GB' : GO"; .-.GD'- : OV-QV :: GR' : GO'-GR\ or GD" : OQ.OQ' :: GR^ : OR. OR'. Similarly, if Oqq' be any other chord through 0, and Gd the parallel semi-diameter, Gd" : Oq.Oq' :: GR"- : OR. OR'; .-.OQ.OQ' -.Oq.Oq' :: GD^ : Gd\ This may otherwise be expressed thus, The ratio of the rectangles of the segments depends only an the directions in which they are draton. The proof is the same if the point be within the ellipse. 93. Pbop. XXXII. If a circle intersect an ellipse in four points, the several pairs of the chords of intersection are equally inclined to the axes. For if QQ', qq' be a pair of the chords of intersection, and if these meet in 0, or be produced to meet in 0, the rect- angles OQ . OQ', Oq . Oq' are proportional to the squares on the parallel diameters. 78 THE ELLIPSE. But these rectangles are equal since QQ', qq' are chords of a circle. Therefore the parallel diameters are equal, and, since equal diameters are equally inclined to the axes, it follows that the chords QQ', qq' are equally inclined to the axes. Conversely, if two chords, not parallel, be equally in- clined to the axes a circle can be drawn through their extremities. For, as in Art. 92, if OQQ', Oqq be two chords, and CD, Cd the parallel semi-diameters, OQ.OQ' : Oq.Oq' :: CD' : Cd' ; but, if CD and Cd be equally inclined to the axes, they are equal, and .■.OQ.OQ'= Oq.Oq', and the points Q, Q', q, q' are concyclic. EXAMPLES. 1. If the tangent at Bmeet the latus rectum produced in D, CDX is a right angle. 2. If PCp be a diameter, and the focal distance pS produced meet the tongent at P in T, SP=8T. 3. If the normal at P meet the axis minor in G' and O'N be the perpendicular from G' on SP, then PN—AC. 4. The tangent at P bisects any straight line perpendicular to AA' and terminated by AP, A'P, produced if necessary. 5. Draw a tangent to an ellipse parallel to a given line. 6. SR being the semi-latus rectum, if RA meet the directrix in E, and SE meet the tangent at Ain T, . AT=AS. 7. Prove that SY : SP :: SR : PG. Find where the angle SPS" is greatest. 8. If two points E and E .be taken in the normal PG such, that PE=P£ = C3, the loci of E and E are circles. EXAMPLES. 79 9. If from the focus S' a line be drawn parallel to SP, it will meet the perpendicular ST in the circumference of a circle. 10. If the normal at P meet the axis major in G, prove that PQ is an harmonic mean between the perpendiculars from the foci on the tangent at P. 11. The straight line NQ is drawn parallel to AP to meet CP in Q ; prove that AQia parallel to the tangent at P. 12. The locus of the intersection with the ordinate of the perpen- dicular from the centre on the tangent is an ellipse. 13. If a rectangle circumscribes an ellipse, its diagonals are the directions of conjugate diameters. 14. If tangents TP, TQ be drawn at the extremities, P, Q of any focal chord of an ellipse, prove that the angle PTQ is half the supple- ment of the angle which PQ subtends at the other focus. 15. If J", ^ be the feet of the perpendiculars from the foci on the tangent at P ; prove that Y, N, Z, C are concyolic. 16. If .4 § be drawn from one of the vertices perpendicular to the tangent at any point P, prove that the locus of th» point of intersection of PS and QA produced will be a circle. 17. The straight lines joining each focus to the foot of the perpen- dicular from the other focus on the tangent at any point meet on the normal at the point and bisect it. 18. If two circles touch each other internally, the locus of the centres of circles touching both is an ellipse whose foci are the centres of the given circles. 19. The subnormal at any point P is a third proportional to the intercept of the tangent at P on the major axis and half the minor axis. 20. If the normal at P meet the axis major in G and the axis minor in g, Gg : Sg :: SA : AX, and if the tangent meet the axis minor in t, St : tg v.BC: CD. 21. If the normal at a point P meet the axis in G, and the tangent at P meet the axis in T, prove that Tq-. TP :: BC : PG, Q being the point where the ordinate at P meets the auxiliary circle. 22. If the tangent at any point P meet the tangent at the extre- mities of the axis AA' in F and F', prove that the rectangle AF, A'F' is equal to the square on the semiaxis minor. 23. TP, TQ are tangents ; prove that a circle can be described with T as centre so as to touch SP, HP, SQ, and HQ, or these lines produced, S and H being the foci. 80 EXAMPLES. 24. If two equal and siuiilar ellipses have the same centre, their points of intersection are at the extremities of diameters at right angles to one another. 25. The external angle between any two tangents to an elUpse is equal to the semi-svuu of the angles which the chord joining the points of contact subtends at the foci. 26. The tangent at any point P meets the axes in T and t; if (S be a focus the angles PSt, STP are equal. 27. A conic is drawn touching an ellipse at the extremities A, B of the axes, and passing through the centre C of the ellipse ; prove that the tangent at C is parallel to AB. 28. The tangent at any point P is cut by any two conjugate diameters in T, t, and the points T, t are joined with the foci S, H respectively; prove that the triangles 8PT, HPt are similar to each other. 29. If the diameter conjugate to CP meet SP, and HP (or these produced) in E and E', prove that SE is equal to HE', and that the circles which circumficribe the triangles SCE, HCE\ are equal to one another. 30. PG is a normal, terminating in the major axis ; the circle, of which PG is a diameter, cuts SP, HP, in K, L, respectively : prove that KL is bisected by PG, and is perpendicular to it. 31. Tangents are drawn from any point in a circle through the foci, prove that the lines bisecting the angles between the several pairs of tangents all pass through a fixed point. 32. If a quadrilateral circumscribe an ellipse, the angles subtended by opposite sides at one of the foci are together equal to two right angles. 33. If the normal at P meet the axis minor in G, and if the tangent at P meet the tangent at the vertex A in V, shew that SG : SC :: PV : VA. 34. P, Q are points in two confocal ellipses, at which the line joining the common foci subtends equal angles ; prove that the tangents at P, Q are inclined at an angle which is equal to the angle subtended by PQ at either focus. 35. The transverse axis is the greatest and the conjugate axis the least of all the diameters. 36. Prove that the locus of the centre of the circle inscribed in the triangle SPS' is an ellipse. EXAMPLES. 81 37. If the tangent and ordinate at P meet the transverse axis in T and N, prove that any circle passing through N and T will cut the auxiUary circle orthogonally. 38. If SY, S' Y' be the perpendiculars from the foci on the tangent at a point P, and PN the ordinate, prove that PY : PT :: NY : NT. 39. If a circle, passing through T and Z, touch the major axis in Q, and that diameter of the circle, which passes through Q, meet the tangent in P, then PQ=BC. 40. From the centre of two concentric circles a straight line is drawn to cut them in P and Q ; from P and Q straight lines are drawn parallel to two given lines at right angles. Shew that the locus of their point of intersection is an ellipse. 41. From any two points P, Q on an ellipse four lines are drawn to the foci S, S' : prove that SP . S'Q and SQ . S'P are to one another as the squares of the perpendiculars from a focus on the tangents at P and Q. 42. Two conjugate diameters are cut by the tangent at any point P inM, N ; prove that the area of the triangle CPM varies inversely as that of the triangle CPN. 43. If P be any point on the curve, and ^ F be drawn parallel to PC to meet the conjugate CD in V, prove that the areas of the triangles CA V, CPN are equal, PN being the ordinate. 44. Two tangents to an ellipse intersect at right angles ; prove that the sum of the squares on the chords intercepted on them by the auxiUary circle is constant. 45. Prove that the distance between the two points on the cir- cumference, at which a given chord, not passing through the centre, subtends the greatest and least angles, is equal to the diameter which bisects that chord. 46. The tangent at P intersects a fixed tangent in T; if aS" is the focus and a line be drawn through S perpendicular to ST, meeting the tangent at P in Q, shew that the locus of Q is a straight line touching the ellipse. 47. Shew that, if the distance between the foci be greater than the length of the axis minor, there will be four positions of the tangent, for which the area of the triangle, included between it and the straight lines drawn from the centre of the curve to the feet of the perpen- diculars from the foci on the tangent, wiU be the greatest possible. 48. Two ellipses whose axes are equal, each to each, are placed in the same plane with their centres coincident, and axes inclined to each other. Draw their common tangents. 49. An ellipse is inscribed in a triangle, having one focus at the orthocentre ; prove that the centre of the ellipse is the centre of the Tl n C g 82 EXAMPLES. nine-point circle of the triangle and that its transverse axis is equal to the radius of that circle. 50. The tangent at any point P of a circle meets the tangent at a fixed point A in T, and T is joined with B the extremity of the diameter passing through A ; the locus of the point of intersection of AP, £T ia an ellipse. 51. The ordinate JfP at a point P meets, when produced, the circle on the major axis in Q. If iS be a focus of the ellipse, prove that SQ : SP :: the axis major : the chord of the circle through § and S, and that the diameter of the ellipse parallel to SP is equal to the same chord. 52. If the perpendicular from the centre C on the tangent at P meet the focal distance SP produced in Ji, the locus of ii is a circle, the diameter of which is equal to the axis major. 53. A perfectly elastic billiard ball lies on an elliptical billiard table, and is projected in any direction along the table : shew that all the lines in which it moves after each successive impact touch an ellipse or an hyperbola confocal with the billiard table. 54. Shew that a circle can be drawn through the foci and the intersections of any tangent with the tangents at the vertices. 55. If CP, CD be conjugate semi-diameters, and a rectangle be described so as to have PD for a diagonal and its sides parallel to the axes, the other angular points will be situated on two fixed straight lines passing through the centre C. 56. If the tangent at P meet the minor axis in T, prove that the areas of the triangles SPS, STS' are in the ratio of the squares on CD and ST. 57. Find the loctis of the centre of the circle tonching the trans- verse axis, SP, and S'P produced. 58. In an ellipse SQ and SQ, drawn perpendicularly to a pair of conjugate diameters, intersect in ^; prove that the locus of Q is a con- centric ellipse. 59. If the ordinate iVP meet the auxiliary circle in Q, the perpen- dicular from S on the tangent at Q is equal to SP. 60. If PT, QT be tangents at corresponding points of an ellipse and its auxiliary circle, shew that PT : QT :: BC : PF. 61. If CQ be conjugate to the normal at P, then is CP conjugate to the normal at Q. 62. PQ is one side of a parallelogram described about an ellipse, having its sides parallel to conjugate diameters, and the lines joining P, § to the foci intersect in D, E; prove that the points D, E and the foci are concyclic. EXAHPLES. 83 63. If the centre, a tangent, and the transverse axis be given, prove that the directrices pass each through a fixed point. 64. The straight line joining the feet of perpendiculars from the focus on two tangents is at right angles to the line joining the intersec- tion of the tangents with the other focus. 65. A circle passes through a focus, has its centre on the major axis of the ellipse, and touches the ellipse : shew that the straight hne from the focus to the point of contact is equal to the latus rectum. 66. Prove that the perimeter of the quadrilateral formed by the tangent, the perpendiculars from the foci, and the transverse axis, will be the greatest possible when the focal distances of the point of contact are at right angles to each other. 67. Given a focus, the length of the transverse axis, and that the second focus lies on a straight line, prove that the ellipse will touch two fixed parabolas having the given focus for focus. 68. Tangents are drawn from a point on one of the equiconjugate diameters ; prove that the point, the centre, and the two points of con- tact are concyclic. 69. If PN be the ordinate of P, and if with centre C and radius equal to PN a circle be described intersecting PN in Q, prove that the locus of Q is an ellipse. 70. If AQO be drawn parallel to CP, meeting the curve in Q and the minor axis in 0, ^CP^=AO . AQ. 71. PS is a focal distance ; CR is a radius of the auxiliary circle parallel to PS, and drawn in the direction from P to S; SQ is a per- pendicular on CR : shew that the rectangle contained by SP and QR is equal to the square on half the minor axis. 72. If a focus be joined with the point where the tangent at the nearer vertex intersects any other tangent, and perpendiculars be let fall from the other focus on the joining line and on the last- mentioned tangent, prove that the distance between the feet of these perpendiculars is equal to the distance from either focus to the remoter vertex. 73. A parallelogram is described about an ellipse; if two of its angular points lie on the directrices, the other two will lie on the auxiliary circle. 74. From a point in the auxiliary circle straight lines are drawn touching the ellipse in P and P' ; prove that SP is parallel to SP. 75. Find the locus of the points of contact of tangents to a series of confooal ellipses from a fixea point in the axis major. 76. A series of confocal ellipses intersect a given straight hne; prove that the locus of the points of intersection of the pairs of tangents drawn at the extremities of the chords of intersection is a straight line at right angles to the given straight line. 6—2 84 EXAMPLES. 77. Given a focus and the length of the major axis ; describe an ellipse touching a given straight hne and passing through a given point. 78. Given a focus and the length of the major axis; describe an ellipse touching two given straight lines. 79. Find the positions of the foci and directrices of an ellipse ■which touches at two given points P, Q, two given straight lines PO, QO, and has one focus on the line PQ, the angle POQ being less than a right angle. 80. Through any point P of an ellipse are drawn straight lines APQ, A'PR, meeting the auxiliary circle in Q, R, and ordinates Qq, Rr are drawn to the transverse axis ; prove that, L being an extremity of the latus rectum, Aq . A'r : At . A'q :: AC^ : SL\ 81. If a tangent at a point P meet the major axis in T, and the perpendiculars from the focus and centre in T and Z, then TT^ :PY^:: TZ : PZ. 82. An ellipse slides between two lines at right angles to each other ; find the locus of its centre. 83. TP, TQ are two tangents, and CP, CQ! are the radii from the centre respectively parallel to these tangente, prove that P^ is parallel toP§. 84. The tangent at P meets the minor axis in t ; prove that St . PN=BC . CD. 85. If the circle, centre t, and radius tS, meet the ellipse in Q, and QM be the ordinate, prove that QM : PN :: BC : BC+CD. 86. Perpendiculars ST, S'T' are let fall from the foci upon a pair of tangents TT, TV; prove that the angles STT, S'TY' are equal to the angles at the base of the triangle TCY'. 87. PQ is the chord of an ellipse normal at P, LCL' the diameter bisecting it, shew that PQ bisects the angle LPL' and that LP+PL' is constant. 88. ABC is an isosceles triangle of which the side AB is equal to the side AG. BD, BE drawn on opposite sides of BC and equally inclined to it meet AC in D and E. If an ellipse is described round BDE having its axis minor parallel to BC, then AB will be a tangent to the ellipse. 89. If A be the extremity of the major axis and P any point on the curve, the bisectors of the angles PSA, PS' A meet on the tangent at P. EXAMPLES. 85 90. If two ellipses intersect in four points, the diameters parallel to a pair of the chords of intersection are in the same ratio to each other. 91. From any point P of an ellipse a straight line PQ is drawn perpendicular to the focal distance SP, and meeting in Q the diameter conjugate to that through P; shew that PQ varies inversely as the ordinate of P. 92. If a tangent to an ellipse intersect at right angles a tangent to a confocal ellipse, the point of intersection lies on a fixed circle. 93. If from a point T in the director circle of an ellipse tangents TP, TP are drawn, the line joining T with the intersection of the normals at P and P' passes through C. 94. Through the middle point of a focal chord a straight line is drawn at right angles to it to meet the axis in R ; prove that SR bears to SC the duplicate ratio of the chord to the diameter parallel to it, . 107. A chord PQ, normal at P, meets the directrices in K and Z, and the tangents at P and Q meet in T; prove that PK and QL subtend equal angles at T, and that KL subtends at 7* an angle which is half the sum of the angles sabtended by Si^ at the ends of the chord. 108. The tangent at the point P meets the directrices in E and F; prove that the other tangents from I! and F intersect on the normal at P. 109. If the tangent at any point meets a pair of conjugate diameters in T and T', prove that TT' subtends supplementary angles at the foci. 110. PSQ, PS'R are focal chords; prove that the tangent at P and the chord QR cut the major axis at equal distances from the centre. CHAPTER IV. The Hyperbola. DEFINITION. An hyperbola is the curve traced hy a point which moves in such a manner, that its distance from a given point is in a constant ratio of greater inequality to its distance from a given straight line. 94. vertex. Troidng the Curve. Let S be the focus, EX the directrix, and A the Then, as in Art. 1, any number of points on the curve may be obtained by taking successive positions of E on the directrix. 88 THE HTPEBBOLA. In SX produced, find a point A' such that SA' : A'X :: 8A : AX. then A' is the other vertex as in the ellipse, and, the eccentricity being greater than unity, the points A and A' are evidently on opposite sides of the directrix. Find the point P corresponding to E, and let A'E, PS produced meet in P', then, if P'K' perpendicular to the directrix meet SE produced in L', FT : P'K' :: SA' : A'X :: SA : AX, and the angle F'L'S = L'SX = L'SP'; .: 8P' = P'L'. Hence P' is a point in the curve, and PSP' is a focal chord. Following out the construction we observe that, since SA is greater than AX, there are two points on the directrix, e and e^, such that Ae and J.e' are each equal to AS. If E coincide with e, the angle QSL = LSI}' = ASe = AeS. Hence SQ, AP are parallel, and the corresponding point of the curve is at an infinite distance ; and similarly the curve tends to infinity in the direction Ae'. Further, the angle ASE is less or greater than AES, according as the point E is, or is not, between e and e'. THE HYPERBOLA. 89 Hence, when E is below e, the curve lies above the axis, to the right of the directrix ; when between e and X, below the axis to the left ; when between X and e', above the axis to the left ; and when above e', below the axis to the right. Hence a general idea can be obtained of the form of the curve, tending to infinity in four directions, as in the figure of Art. 102. DEFINITIONS. The line A A' is called the transverse awis of the hyperbola. The middle point, G, of A A' is the centre. Any straight line, drawn through G and terminated by the curve, is called a diameter. 95. Prop. I. If P be any point of aii hyperbola, and AA' its transverse aons, and if A'P, and PA produced, {or PA and PA' produced) meet the directrix in E and F, EF subtends a right angle at the focus. By the theorem of Art. 4, ES bisects the angle ASP' and i^/Sf bisects ^/SfP; .". ESF is a right angle. SAA' being a focal chord, this is a particular case of the theorem of Art. 6. 96. Prop. II. If PN be the ordinate of a point P, and AG A' the transverse axis, PJV" is to AN .NA' in a constant ratio. 90 THE HTFEBBOLA. Join AP, A'P, meeting the directrix in E and F. Then PN : AN : and PN : A'N : .-. PN' : AN.NA' : EX : AX, FX : A'X; EX.FX : AX. A'X SX' : AX. A'X, since ESF is a right angle; that is, PN^ is to AN.NA', in a constant ratio. Through C, the middle point of AA', draw CB at right angles to the axis, and such that BC : AC :: 82^ : AX. A'X; then PN* : AN.NA' :: BC : AC, or PN' : CN'-AC :: BC : AC. CoE. If PM be the perpendicular from P on BG, PM= ON, and PN=CM; .-. CM' : PM'-AC :: BC : AC, or CM' : 5(7' :: PM' - AC : AC .: CM' + BC : BC :: PJT : AC or PJIf' : CM' + BC :: AC : BC. 97. If we describe the circle on AA' as diameter, which we may term, for convenience, the aiuciliary circle, the rectangle AN . NA' is equal to the square on the tangent to the circle from N. THE HTPEBBOLA. 91 Hence the preceding theorem may be thus expressed : The ordinate of an hyperbola is to the tangent from its foot to the auxiliary circle in the ratio of the conjugate to the transverse axis. Def. If GBf he taken equal to CB, on the other side of the axis, the line BCB' is called the conjugate oms. The two lines AA', BB' are the principal axes of the curve. When these lines are equal, the hyperbola is said to be equilateral, or rectangular. The lines AA', BB" are sometimes called major and minor axes, but, as AA' is not necessarily greater than BB", these terms cannot with propriety be generally employed. If a point JV' be taken on CA' produced, such that CJV' = CN, the corresponding ordinate P'N' = Plf, and therefore it foUoTTS that the curve is symmetrical with regard to BCB', and that there is another focus and directrix, corresponding to the vertex A'. 98. Pbop. IIL If ACA' be the transverse axis, C the centre, S one of the foci, and X the foot of the directrix, CS : GA :: GA : GX :: SA : AX, and GS : GX :: GS' : GA\ Interchanging the positions of S and X for a new S' A' X' G X A S figure, the proof of these relations is identical with the proof given for the ellipse in Art. 61. 99. Prop. IV. If S be a focus, and B an extremity of the conjugate axis, BG' = AS. 8A', &ndSG' = AG' + BC 92 THE HYFEBBOLA. Referring to Art. (98), SX = SA + AX ; .-.SX : AX':: SA + AX : AX, :: SC+AG : AG; and similarly SX : A'X :: SG-AG : AG; .-.SX* : AX. A'X :: SG'-AG^ : AG'. . But BG' : AG' :: SX' : AX. A'X; .■.BG'=SC'-AG' = AS.SA'. Hence SG' = AG' + BG' = AB' ; i.e. SG is equal to the line joining the ends of the axes. 100. Prop. V. The difference of the focal distances of any point is equal to the transverse axis. For, if PKK', perpendicular to the directrices, meet them in K and K' , S'P : PK' :: SA : AX, and SP : PK :: SA : AX ; .-.S'P-SP : KK' :: SA : AX, :: AA' : XX' (Art. 98) ; .■.S'P-SP = AA'. COK. 1. SP : NX :: AG : GX; -■. SP : AG :: NX : GX; .-.SP + AG : AG :: GN : GX. or SP + AG : GN :: SA : AX. Hence also S'P -AG : GN :: SA : AX. Cor. 2. Hence also it can be easily shewn, that the difference of the distances of any point from the foci of an hyperbola, is greater or less than the transverse axis, according as the point is within or without the concave side of the curve. THE HYPERBOLA. 93 101. Mechanical Construction of the Hyperbola. Let a straight rod 8'L be moveable in the plane of the paper about the point S'. Take a piece of string, the length of which is less than that of the rod, and fasten one end to a fixed point S, and the other end to L; then, pressing a pencil against the string so as to keep it stretched, and a part of it PL in contact with the rod, the pencil will trace out on the paper an hyperbola, having its foci at 8 and S', and its transverse axis equal to the difference between the length of the rod and that of the string. This construction gives the right-hand branch of the curve ; to trace the other branch, take the string longer than the rod, and such that it exceeds the length of the rod by the transverse axis. We may remark that by taking a longer rod MS'L, and taking the string longer than SS' + S'L, so that the point P will be always on the end S'M of the rod, we shall obtain an ellipse of which 8 and 8' are the foci. Moreover, re- membering that a parabola is the limiting form of an ellipse when one of the foci is removed to an infinite distance, the mechanical construction given for the parabola will be seen to be a particular case of the above. The Asymptotes. 102. We have shewn in Art. 94 that if two points, e and e', be taken on the directrix such that Ae = Ae' = A8, the lines eA, e'A meet the curve at an infinite distance. 94 THE HYPERBOLA. These lines axe parallel to the diagonals of the rectangle formed by the axes, for Ae' : AX :: AS : AX :: SG : AC, :: AB : AC, (Art. 99). Definition. The diagonals of the rectangle formed by the principal axes are called the asymptotes. We observe that the axes bisect the angles between the asymptotes, and that if a double ordinate, PNP', when produced, meet the asymptotes in Q and Q, PQ=Fq. The figure appended will give the general form of the curve and its connection with the asymptotes and the auxiliary circle. 103. Pbop, VI. The asymptotes intersect the directrices in the same points as the aumliary circle, and the lines joining the corresponding foci with the points of intersection are tangents to the circle. If the asymptote CL meet the directrix in D, ioining SB (fig. Art 102), CL'' = AC + BC*=8a', and CD : CX :: CL : GA :: SC : CA :: CA : GX ; .'. CD=CA, and D is on the auxiliary circle. THE HYPERBOLA. 95 Also CS.GX = CA'' = CI>'; .-. CDS is a right angle, and SD is the tangent at D. Cor. GB' + 8D' = CS' = AG' + BG* (Art. 99) ; .-. SD = BC. 104. An asymptote may also be characterized as the ultimate position of a tangent when the point of contact is removed to an infinite distance. It appears from Art. 10 that in order to find the point of contact of a tangent drawn from a point J" in the directrix, we must join T with the focus S, and draw tm-ough S a straight line at right angles to ST; this line will meet the curve in the point of contact. In the figures of Arts. 94 and 102 we know that the line through S, parallel to eA or CL, meets the curve in a point at an infinite distance, and also that this straight line is at right angles to SD, since SD is at right angles to CD. Hence the tangent from D, that is the line from D to the point at an infinite distance, is perpendicular to DS a,nd there- fore coincident with CD. The asymptotes therefore touch the curve at an infinite distance. 105. Def. If an hyperbola be described, having for its transverse and conjugate axes, respectively, the conjugate and transverse axes of a given hyperbola, it is called the conjugate hyperbola. It is evident from the preceding article that the conju- gate hyperbola has the same asjrmptotes as the original hyperbola, and that the distances of its foci from the centre are also the same. The relations of Art. 96 and its Corollary are also true, mutatis mutandis, of the conjugate hyperbola ; thus, if R be a point in the conjugate hyperbola, RM^ : GM^-BC :: AC^ : BC, aiid CW-.RM^ + AG^-.-.BG'-.AG'. Def. a straight line drawn through the centre and ter- minated by the conjugate hyperbola is also called a diameter of the original hyperbola. 96 THE HYPERBOLA. 106. Prop. VII. If from, any point Q in one of the asymptotes, two straight lines QP^, QRM he drawn at right angles respectively to the transverse and conjugate axes, and meeting the hyperbola in P, p, and the conjugate hyperbola in R, r, QP.Qp = BG\ and QR .Qr= AC- s^q^ or I.e. For QN' : BO' :: ON' : AC ; .: QN'-BC :BC':: GIP-AC' : AC •.-.PN'-.BC'; .■.Q]!P-BC = PN\ Q2f''-PN'' = BC'; QP.Qp = BC. Similarly, QM' : AC :: CM' : BC; .-. QM'-AC : AC :: CUP-BC : BC, ■.-.RAP: AC; .■.QM'-RM' = AC, or qR.Qr = AC. These relations may also be given in the form, QP.Pq = BC, QR.Rq' = AC. Cor. If the point Q be taken at a greater distance from G, the length QN and therefore Qp will be increased, and may be increased indefinitely. THE HYPERBOLA. 97 But the rectangle QP . Qp is of finite magnitude ; hence QP will be indefinitely diminished, and the curve, therefore, as it recedes from the centre, tends more and more nearly to coincide with the asymptote. A further illustration is thus given of the remarks in Art. 104. 107. If in the preceding figure the line Qq be produced to meet the conjugate hyperbola in E and e, it can be shewn, in the same manner as in Art. 106, that QE.Qe = BC; and this equality is still true when the line Qq lies between C and A, in which case Qq does not meet the hyperbola. Properties of the Tangent and Normal. 108. In the case of the hyperbola the theorem, proofs of which are given in Arts. 15 and 16, takes the following form: The tangents drawn from any point to an hyperbola subtend equal or supplementary angles at either focus ax- cording as they tovch the same or opposite branches of the curve. For, T being the point of intersection of tangents to opposite branches of the curve, let TM, TM' be the per- B. c. s. 7 98 THE UYPERBOLA. pendiculars let fall from T on BF and /SQ, then, as in Arts. 15 andl6, 2'Jlf=2W; .•. the angles TBM, TSM are equal, and consequently the angles TSP, TSQ are supplementary. 109. Prop. VIII. The tangent at any point Insects the angle between the focal distances of that point, and the normal is equally inclin^ to the focal distances. Let the normal at P meet the axis in G. Then (Art. 18), 8G : 8P :: SA : AX, and S'G:S'P::8A:AX; .: SG:S'G::SP:8'P] and therefore the angle between 8P and 8'P produced is bisected by PG. Hence PT, the tangent which is perpendicular to PG, bisects the angle 8PS'. Cor. 1. If PT and GP produced meet, respectively, the conjugate axis in t and g, it can be shew^n, in exactly the same manner as in the corresponding case of the ellipse (Art. 68), that 8, P, 8', t, and g are concyclic. Cor. 2. If an ellipse be described having 8 and 8' for its foci, and if this eUipse meet the hyperbola in P, the normal at P to the ellipse bisects the angle 8P8', and there- fore coincides with the tangent to the hyperbola. Hence, if an ellipse and an hyperbola be confocal, that is, have the same foci, they intersect at right angles. 110. Prop. IX. Every diameter is bisected at the centre, and the tangents at the ends of a diameter are parallel. Let PCp be a diameter, and PN, pn the ordinates. Then CJP : Cn" :: PIP : pn", ■.■.CN'- AC: On' -AC; hence CN = Gn, and .-. GP = Op. THE HYPERBOLA. Again, if FT, pt be the tangents, 99 The triangles PCS, pGS' are equal in all respects, and therefore SPS'p is a parallelogram. Hence the angles SPS', SpS' are equal, and therefore SPT=S'pt. But SPC=S'pC, .•. the difference TPC = the difference tpG, and PT is parallel to pt. It can be shewn in exactly the same manner, that, if the diameter be terminated by the conjugate hyperbola, it is bisected in G, and the tangents at its extremities are parallel. COK. The distances SP, Sp are equally inclined to the tangents at P and p. 111. Prop. X. The perpendicvlars from the fod on any tangent meet the tangent on the auxiliary circle, and the semi-conjugate axis is a mean proportional between their lengths. Let 8Y, S'Y be the perpendiculars, and let (SF produced meet S'P in L. Then the triangles SPY, LPT are equal in all respects. and SY = LY. 1—2 100 THE HYPERBOLA. Hence, being the middle point of SS' and Y of SL, CT is parallel to S'L, and S'L = 2CY. But 8'L = S'P-PL = S'P-SP = 2AC; .■.CY=AG. and Y is on the auxiliary circle. So ako y is a point in the circle. Let 8Y produced meet the circle in Z, and join Y'Z; then, Y'YZ being a right angle, ZY' is a diameter and passes through G. Hence, the triangles SCZ, S'CY being equal, J^Y' = SZ, and SY. S'Y' = SY.SZ = SA. SA' = BC. CoK. 1. If P" be the other extremity of the diameter PC, the tangent at P" is parallel to PY, and therefore Z is the foot of the perpendicular from 8 on the tangent atP'. Cor. 2. If the diameter BCD', drawn parallel to the tangent at P, meet S'P, SP in E and E', PEGY is a parallelogram ; .■.PE=CY = AO, and so also PE' =CY' = A G. THE HYPERBOLA. 101 112. Prop. XI. To draw tangents to an hyperbola from a given point. The construction of Art. 17 may be employed, or, as in the cases of the ellipse and parabola, the following. Let Q be the given point; join SQ, and upon SQ as diameter describe a circle intersecting the auxiliary circle in FandT'; QFand QY' are the required tangents. Producing SY to L, so that YL = SY, draw S'L cutting QY in P, and join SP. The triangles SPY, LPY are equal in all respects, and S'P-SP = S'L=20Y=2AG; .*. P is a point on the hyperbola. Also QP bisects the angle SPS', and is therefore the tangent at P. A similar construction will give the other tangent QP'. If the point Q be within the angle formed by the asymp- totes, the tangents will both touch the same branch of the curve ; but if it lie within the external angle, they will touch opposite branches. 102 THE HYPERBOLA. 113. Peop. XII. If two tangents be drawn from any point to an hyperbola they are equally inclined to the focal distances of that point. Let PQ, P'Q be the tangents, SY, S'T, SZ, S'Z' the perpendiculars from the foci ; join TZ, Y'Z'. Then the angles YSZ, Y'S'Z' are equal, for they are the supplements of YQZ, Y'QZ'. Also 8Y.8'Y'=SZ. S'Z' (Ari 111) ; or SY : SZ :: S'Z' : S'Y' ; .: the triangles YZS, Y'S'Z' are similar, and the angle YZS = Z'Y'S'. But the angle YQS= YZS, and Z'QS' = ZY'S' ; .-. YQS=Z'Q8'. That is, the tangent QP and the tangent P'Q produced are equally inclined to 8Q and S'Q. Or, producing S'Q, QP and QP' are equally inclined to Q8 and S'Q produced. In exactly the same manner it can be shewn that if QP, QP' touch opposite branches of the curve the angles PQS, P'QS' are equal. Cob. If Q be a point in a confocal hyperbola, the nor- mal at Q bisects the angle between SQ and S'Q produced and therefore bisects the angle PQP". THE HYPERBOLA. 103 Hence, if from any point of an hyperbola tangents be drawn to a confocal hyperbola, these tangents are equally inclined to the normal or the tangent at the point, according as it lies within or without that angle formed by the asymptotes of the confocal which contains the transverse axes. 114. Prop. XIII. If FT, the tangent at P, meet the transverse axis in T, and FN be the ordinate, CF.CT = AG\ Let fall the perpendicular SY upon FT, and join FiV, CY, SF, and S'F. The angle GYT=S'FY = SFY = the supplement of SNY= CNY ; eJso the angle YGT is com- mon to the two triangles CYT, CYN; these triangles are therefore similar, and or CN : CY :: GY : GT, GN.GT=^GT' = AG\ Cor. 1. JLeuc&GN.NT ^CN'-GN.CT = CN'-AG' = AN. If a: Cor. 2. Hence also it follows that If any number of hyperbolas be described having the same transverse axis, and an ordinate be drawn cutting the hyperbolas, the tangents at the points of section will all meet the transverse axis in the same point. Cor. 3. If CN be increased indefinitely, CT is dimi- nished indefinitely, and the tangent ultimately passes through C, as we have already shewn in Art. 104. 104 THE HTPEKBOLA. 115. Prop. XIV. If the tangent at P meet the conjugate axis in t, and PN he the ordinate. For Gt.PN = BC\ Ct : PN :: GT : NT; (Fig. Art. 114) Gt . PN : PN^ : .Gt.PN : AG* : and GT.GN : GN.NT AC : AN.NA'. PN' : AN.NA' BG* : AG\ Gt.PN = BG\ In exactly the same manner as in Art. 76, it can be shewn that GG.GT = 8G\ GG : GN :: SG* : AG\ Gg : PN :: 8G* : BG\ and NG : GN :: BG* : AG\ 116. Pbop. XV. If the normal at P meet the transverse axis in G, the conjugate axis in g, and the diameter parallel to the tangent at P in F, PF.PG = BG', and PF.Pg= AG*. Let NP, PM, perpendicular to the axes, meet the diameter Cf in £■ and i ; Then KNG, KFG being right angles, K, F, N, Q are concyclic ; ^ .:PF.PO=PK.PN = Gt.PN=BG\ Similarly F, L, M, g are con- cyclic ; .-. PF. Pg=PL. PM= GT.GN=AG THE HYPERBOLA. 105 117. Prop. XVI. If PCp he a diameter, and QV an ordinate, and if the tangent at Q meet the diameter Pp in T, CV.CT=CP\ Let the tangents at P and p meet the tangent at Q in Jt and r ; Then the angle SPR = Spr (Cor. Art. 110) and therefore if RN", m be the perpendiculars on SP, sp, the triangles ItPN, rpn are similar. Draw RM, rm perpendiculars on SQ. Then TR : Tr :: RP : rp :: RN : m, •.-.RM-.rm (Cor. Art. 15) :: RQ : rQ. Hence, QV, RP, and rp being parallel, TP -.Tp-.-.PV-.pV; .: TP + Tp:Tp-TP.: PV+pV : pV-PV, or 2CP : 2CT :: 2CF : 2CP, or CV.CT=CF'. 106 THE HYPERBOLA. 118. Prop. XVII. A diameter bisects all chords par- allel to the tangents at its extremities. Let PCp be the diameter, and QQ' the chord, parallel to the tangents at P and p. Then if the tangents TQ, TQ' at Q and Q' meet the tangents at P and p, in the points £, E', e, e', EP = E'P and ep = e'p, (Art. 21) .". the point T is on the line Pp ; but TP bisects QQ' ; that is, the diameter pCP produced bisects QQ. Def. The line DCd, drawn parallel to the tangent at P and terminated by the conjugate hyperbola, that is, the diameter parallel to the tangent at P, is said to be conjugate to PCp. A diameter therefore bisects all chords parallel to its conjugate. 119. Prop. XVIIL If the diamMer DCd he conjugate to PCp, then will PCp he conjugate to DCd. Let the chord QFg be parallel to CD and be bisected in V by CP produced. Draw the diameter qCR, and join RQ meeting CD in U. THE HYPERBOLA. 107 Then RG== Gq and Qr=rq; :. QR is parallel to OP. Also QU : UR :: Cq : CR, and .-. QU= UR, that is, CD bisects the chords parallel to CP, and PCp is therefore conjugate to BGd. Hence, when two diameters are conjugate, each bisects the chords parallel to the other. Def. Ghords drawn from, the extremities of any diameter to a point on the hyperbola are called supplemental chords. Thus, qQ, QR are supplemental chords, and they are parallel to GD and GP ; supplemental chords are therefore parallel to conjugate diameters. Def. a line QV, drawn from any point Q of an hyperbola, parallel to a diameter DGd, and terminated by the conjugate diameter PGp, is called an ordinate of the diameter PCp, and if QV produced meet the curve in Q', QVQ' is the double ordinate. This definition includes the two cases in which QQ' may- be drawn so as to meet the same, or opposite branches of the hyperbola. 120. Peop. XIX. Any diameter is a mean proportional between the transverse axis and the focal chord parallel to the diameter. This can be proved as in Art. 81. 108 THE HTPEEBOLA. Properties of Asymptotes. 121. Prop. XX. I/from any point Q in an asymptote QPpq be drawn meeting the curve in P, p and the other asymptote in q, and if CD be the semi-diameter parallel to Qq, QP .Pq= CB^ and QP =pq. Through P and D draw RPr, BTt perpendicular to the transverse axis, and meeting the asymptotes. Then QP : BP :: CD : DT. and Pq:Pr::CD:Dt; .: QP.Pq-.EP. Pr :: GB' : DT . Dt. But RP .Pr = BC- = DT . Dt (Arts. 106 and 107), :. QP . Pq= CB'. Similarly qp . pQ = CD" ; .■.QP.Pq = qp.pQ; or, if V he the middle point of Qq, QV*-PV' = QV^-pY\ Hence PV =pV, dcoA :.PQ=pq. We have taken the case in which Qq meets one branch of the hyperbola. It may however be shewn in the same manner that the same relations hold good for the case in which Qq meets opposite branches. THE HYPERBOLA. 109 Cor. If a straight line PP'p'p meet the hyperbola in P, p, and the conjugate hyperbola in P', p', PP" =pp'. For, if the line meet the asymptotes in Q, q, QP=p'q,B.nAPQ = qp; :. PP'=pp'. 122. Prop. XXI. The portion of a tangent which is terminated by the asymptotes is bisected at the point of contact, and is equal to the parallel diameter. LEI being the tangent (Fig. Art. 121), and DCd the parallel diameter, draw any parallel straight line QPpq meeting the curve and the asymptotes. Then QP=pq; and, if the line move parallel to itself until it coincides with LI, the points P and p coincide with E,a,nd.:LE = M. Also QP .Pq = CD', always ; .-. LE . El = GD\ or LE= CD. Properties of Conjugate Diameters. 123. Prop. XXII. Conjugate diameters of an hyperbola are also conjugate diameters of the conjugate hyperbola, and the asymptotes are diagonals of the parallelogram formed by the tangents at their extremities. PCp and DGd being conjugate, let QVq, a double ordinate of CD, meet the conjugate hyperbola in Q' and q'. Then Qr= Yq, and QQ'=qq' (Cor. Art. 121), .:Q^r=Vq'. That is, CD bisects the chords of the conjugate hyperbola parallel to CP. Hence CD and CP are conjugate in both hyperbolas, and therefore the tangent at D is parallel to CP. 110 THE HTPERBOLA. Let the tangent at P meet the asymptote in L ; then PL = GD (Art. 122). Hence LD is parallel and equal to CP ; but the tangent at D is parallel to CP ; .'. LD is the tangent at D. Completing the figure, the tangents at p and d are par- allel to those at P and D, and therefore the asymptotes are the diagonals of the parallelogram Lll'L'. CoE. Hence, joining PD, it follows that PD is parallel to the asymptote IGL', since LP — PL', and LD = Dl. 124. Prop. XXIII. If QV he an ordinate of a dia- TMskr PCp, and DCd the conjugate diameter, QT-.PV. Vpy.CD'-.CP'. Let QV and the tangent at P meet the asymptote in R and L. THE HTPEBBOLA. Ill Then LP being equal to CD, RT-.CD'-.-.CV-.GP'; :. RV- CD' -.CD':: CV - CP' : GP'. But BT-Qr''='CD'. Hence Q V : CD' ■.-.GT-CP': CP', or QV-.PV. Vp :: CD' : GP'. 125. Pkop. XXIV. If QV be an ordinate of a dia- meter PGp, and if the tangent at Q meet the conjugate diamieter, DCd, in t, Ct.Qr= CD'. For, (Fig. Art. 118) Ct : QV :: GT : VT, and .-. Ct. QV: QV :: CV . CT : GV . VT. But CV.CT=GP', andCF. VT = GV'-CV .GT=GV'-GP'; :. Ct.QV: QV :: CP' : CV-CP', :: CD' : QV. Hence Ct.QV =CD^. 126. Prop. XXV. If ACa, BCb he conjugate diameters, and PGp, DCd another pair of conjugate diameters, and if PN, DM be ordinates of ACa, CM : PN :: AG : BC, and DM : GN :: BC : AG. Let the tangents at P and D meet ACa in T and t ; then CN. CT = AC' = CM.Ct (Art. 117), .-. CM : CN :: GT : Ct. :: PT : CD, :: PN : DM. :: GN : Mt ; 112 THE HYPERBOLA. .-. CJV* = CM . Mt = CM' + CM.Gt = CM' + AC, so that CM* = GN' - A C\ But FN* : CN'-AC :: BC : AC*; :. CM : PN :: AC : BC; and, similarly, DM : CN :: BC : AG. Cob. We have shewn in the course of the proof, that CN*-GM* = AG\ Similarly, if Pn, Dm be ordinates of BC, Gm*-Cn* = BG*; that is, I)M*-PN* = BC*; and it must be noticed that these relations are shewn for any pair of conjugate diameters AGa, BGh, including of course the axes. 127. Pkop. XXVI. If GP, CD be conjugcOe semi- diameters, and AC, BC the semi-axes, CP'-CD'=AC*-BC\ For, drawing the ordinates PN, DM, and remembering that in this case the angles at N and M are right angles, we have, from the figure of the previous article, GP' = GN* + PN\ CI)*=GM* + DM\ But CN'-CM' = AG' and DM'-PN*= BC*; .'.CP'-CB' = AC*-BC*. THE HYPERBOLA. 113 128. Phop. XXVII. Jf the normal at P meet the axes in G and g, PG : CD :: BG : AG, and Pg : GB :: AG : BG. For the proofs of these relations, see Art. 86. Observe also that PG.Pg = GD\ and that Gg : CD :: SG' : AC.BC. 129. Prop. XXVIII. The area of the parallelogram formed hy the tangents at the ends of conjugate diameters is equal to the rectangle contained by the axes. Let CP, CD be the semi-diameters, and PN, DM the ordinates of the transverse axis. Let the normal at P meet CD in F, and the axis in G. Then PJSfG, CDM are similar triangles, and, exactly as in Art. 87, it can be shewn that PF.GD = AG.BG. Hence it follows that, in the figure of Art. 123, the tiiangle LGL' is of constant area. For the triangle is equal to the parallelogram GPLD. 130. Prop. XXIX. If SP, S'P he the focal distances of a point P, and CD be conjugate to CP, SP.S'P = GD'. Attending to the figure of Art. Ill, the proof is the same as that of Art. 88. 114 THE HYPERBOLA. 131. Prop. XXX. If the tangent at P meet a pair of conjugate diameters in T and t, and CD he conjugate to GP, PT.Pt=CD\ This can be proved as in Art. 89. It can also be shewn that if the tangent at P meet two parallel tangents in T' and t', PT'.P^ = GD\ 132. Prop. XXXI. If the tangent at P m,eet the asymptotes in L and L', GL.CL' = SG*. Let the tangent at A meet the asymptotes in K and K' ; then (Art. 129) the triangles LGL', KGK' are of equal area, and therefore GL : GK' :: GK : GL' (EucHd, Book vi.), or CL.GL' = GK* = AG'+BG* = SG\ Cor. If PH, PH' be drawn parallel to, and terminated by the asymptotes, J^.PH.PH' = G8\ for GL = tPH', and GL' = 2PH. THE HYPERBOLA. 115 133. Prop. XXXII. Pairs of tangents at right angles to each other intersect on a fixed circle. PT, QT being two tangents at right angles, let SY, perpendicular to PT, meet S'P in K. Then (Art. 113) the angle S'TY' = QTS, and obviously, KTP = PTS ; therefore S'TY' is complementary to KTP, and S'TK is a right angle. Hence 44 G' = S'K' = ST + TK" = S'T' + Sr = 2.Cr+2.CS'' by Euclid ii. 12 and 13 ; .■.Cr = AC*-BG', and the locus of T is a circle. If AG he less than BG, this relation is impossible. In this case, however, the angle between the asymptotes is greater than a right angle, and the angle PTQ between a pair of tangents, being always greater than the angle between the asymptotes, is greater than a right angle. The problem is therefore a priori impossible for the hyperbola, but be- comes possible for the conjugate hyperbola. As in the case of the ellipse, the locus of T is called the director circle. 8—2 116 THE HYPERBOLA. 134. Prop. XXXIII. The rectangles contained by the segments of any two chords which intersect each other are in the ratio of the squares on the parallel diameters. Through any point in a chord QOQ' draw the diameter ORR ; and let CD be parallel to QQ', and CP conjugate to CD, bisecting QQ in V. Draw RU axL ordinate of CP. Then But RU* : CU^-GP' .-.CD'+RU' : CU" CD'' : CP'; CD' : CP', CD' + QV : CV. OV : CV; CD^ RU' : GU' CU' :: CD' + QT-OT : CT. CD' : CD' + QV-OV :: CU^ : CV, :: CR' : 00"; . CD' : QV- OV :: CR : C(f-CR^, CDf : QO.OQ' :: CE' : OR. OR'. Similarly, if qOq' be any other chord, and Cd the parallel semi-diameter, C(P : qO.Oi :: GP^ : OR. OK; .-.QO.OQ' : qO.Oq :: CD' : Cd!'; that is, the ratio of the rectangles depends only on the directions of the chords. or or EXAMPLES. 117 Prop. XXXIV. If a circle intersect an hyperbola in four points, the several pairs of the chords of intersection are equally inclined to the axes. For the proof, see Art. 93. Examples. 1. If a circle be drawn so as to touch two fixed circles externally, the locus of its centre is an hyperbola. 2. If the tangent at B to the conjugate meet the latus rectum in D, the triangles SCD, SXD are similar. 3. The straight line drawn from the focus to the directrix, parallel to an asymptote, is equal to the semi-latus-rectum, and is bisected by the curve. 4. Given the asymptotes and a focus, find the directrix. 5. Given the centre, one asymptote, and a directrix, find the focus. 6. Parabolas are described passing through two fixed points, and having their axes parallel to a fixed line ; the locus of their foci is an hyperbola. 7. The base of a triangle being given, and also the point of contact with the base of the inscribed circle, the locus of the vertex is an hyperbola. 8. If the normal at P meet the conjugate axis in g, and gN be the perpendicular on SP, then PN=AC. 9. Draw a tangent to an hyperbola, or its conjugate, parallel to a given line. 10. If AA' be the axis of an ellipse, and PNP" a double ordinate, the locus of the intersection of A'P and PA is an hyperbola. 11. The tangent at P bisects any straight line perpendicular to AA', and terminated by AP, and A'P. 12. If PCp be a diameter, and if *S^ meet the tangent at P in T, SP^ST. 13. Given an asymptote, the focus, and a point ; construct the hyperbola. 118 EXAMPLES. 14. A circle can be drawn through the foci and the intersections of any tangent with the tangents at the vertices. 15. Given an asymptote, the directrix, and a point ; construct the hyperbola. 16. If through any point of an hyperbola straight lines are drawn parallel to the asymptotes and meeting any semi-diameter CQ in P and R, CP. CR=C^. 17. PJfia an ordinate and ^Q parallel to AB meets the conjugate axis in Q ; prove that QB . Qff=PN^. 18. NP is an ordinate and Q a point in the curve ; AQ, A'Q meet NP in D and E ; prove that ND . NE=NP^. 19. If a tangent cut the major axis in the point T, and perpen- diculars ST, HZ be let fall on it from the foci, then AT.A'T=TT .ZT. 20. In the tangent at P a point Q is taken such that PQ is pro- portional to CD ; shew that the locus of Q is an hyperbola. 21. Tangents are drawn to an hyperbola, and the portion of each tangent intercepted by the asymptotes is divided in a constant ratio ; prove that the locus of the point of section is an hyperbola. 22. If the tangent and normal at P meet the conjugate axis in t and K respectively, prove that a circle can be drawn through the foci and the tlu^e points P, t, K. Shew also that OK : SK :: SA : AX, and St : tK :: BC : CD, CD being conjugate to CP. 23. Shew that the points of trisection of a series of conterminous circular arcs lie on branches of two hyperbolas ; and determine the distance between their centres. 24. If the tangent at any point P cut an asymptote in T, and if SP cut the same asymptote in Q, then SQ=QT. 25. A series of hyperbolas having the same asymptotes is cut by a straight line parallel to one of the asymptotes, and through the points of intersection lines are drawn parallel to the other, and equal to either semi-axis of the corresponding hyperbola : prove that the locus of their extremities is a parabola. 26. Prove that the rectangle PT . PY in an ellipse is equal to the square on the conjugate axis of the confocal hyperbola passing through P. EXAMPLES. 119 27. If the tangent at P meet one asymptote in T, and a line TQ be drawn parallel to the other asymptote to meet the curve in Q ; prove that if PQ be joined and produced both ways to meet the asymptotes in R and R, RR will be trisected at the points P and Q. 28. The tangent at a point P of an elUpse meets the hyperbola having the same axes as the ellipse in C and D. If § be the middle point of CD, prove that OQ and OP are equally inclined to the axes, being the centre of the ellipse. 29. Given one asymptote, the direction of the other, and the position of one focus, determine the position of the vertices. 30. Two points are taken on the same branch of the curve, and on the same side of the axis ; prove that a circle can be drawn touching the four focal distances. 31. Supposing the two asymptotes and one point of the curve to be given in position, shew how to construct the curve ; and find the position of the foci. 32. Given a pair of conjugate diameters, construct the axes. 33. If PH, PK be drawn parallel to the asymptotes from a point P on the curve, and if a line through the centre meet them in R, T, and the parallelogram PRQT be completed, § is a point on the curve. 34. The ordinate NP at any point of an ellipse is produced to a point Q, such that NQ is equal to the subtangent at P ; prove that the locus of § is an hyperbola. 35. If a given point be the focus of any hyperbola, passing through a given point and touching a given straight line, prove that the locus of the other focus is an arc of a fixed hyperbola. 36. An ellipse and hyperbola are described, so that the foci of each are at the extremities of the transverse axis of the other ; prove that the tangents at their points of intersection meet the conjugate axis in points equidistant from the centre. 37. A circle is described about the focus as centre, with a radius eqvial to one-fourth of the latus rectum : prove that the focal distances of the points at which it intersects the hyperbola are parallel to the asymptotes. 38. The tangent at any point forms a triangle with the asymptotes : determine the locus of the point of intersection of the straight lines drawn from the angles of this triangle to bisect the opposite sides. 39. If (ST, (S"y be the perpendiculars on the tangent at P, a circle can be drawn through the points Y, Y', N, C. 40. The straight lines joining each focus to the foot of the per- pendicular from the other focus on the tangent meet on the normal and bisect it. 120 EXAMPLES. 41. If the tangent and normal at P meet the axis in T and ? as diameter, the tangents at the vertices will intersect the asymptotes in the circumference. 10. If two concentric rectangular hyperbolas be described, the axes of one being the asymptotes of the other, they will intersect at right angles. 11. If the tangents at two points Q and Q" meet in T, and if CQ, CQf meet these tangents in R and R', the points R, T, R', are con- cyclic. 12. If from a point Q. in the conjugate axis QA be drawn to the vertex, and §iJ parallel to the transverse axis to meet the curve, QR=A and P be taken in it, so that, PD being a perpendicular upon AB, CD may be of constant magnitude, the locus of /" is a rectangular hyperbola. 15. Every conic passing through the centres of the four circles which touch the sides of a triangle, is a rectangular hyperbola. 16. Ellipses are inscribed in a given parallelogram, shew that their foci lie on a rectangular hyperbola. 17. If two focal chords be parallel to conjugate diameters, the lines joining their extremities intersect on the asymptotes. 18. If P, § be two points of a rectangular hyperbola, centre 0, and QN the perpendicular let faU on the tangent at P, the circle through 0, N, and P wiU pass through the middle point of the chord 19. Having given the centre, a tangent, and a point of a rectangular hyperbola, construct the asymptotes. 20. If a right-angled triangle be inscribed in the curve, the normal at the right angle is parallel to the hypothenuse. 21. On opposite sides of any chord of a rectangular hyperbola are described equal segments of circles ; shew that the four points, in which the circles, to which these segments belong, again meet the hyperbola, are the angular points of a parallelogram. 22. Two lines of given lengths coincide with and move along two fixed lines, in such a manner that a circle can always be drawn through their extremities ; the locus of the centre is a rectangular hyperbola. 134 EXAMPLES. 23. If a rectangular bTperboIa, having its asymptotes coincident with the axes of an ellipse, touch the ellipse, the axis of the hyperbola is a mean proportional between the axes of the ellipse. 24. The tangent at a point P of a rectangular hyperbola meets a diameter QC^ in T. Shew that CQ and TQ subtend equal angles at P. 25. If ^ be any point in a rectangular hyperbola, of which is the centre, BOC the straight line through at right angles to OA, D any other point in the curve, and DS, DC parallel to the asymptotes, prove that B, D, Ai C are concychc. 26. The angle subtended by any chord at the centre is the supple- ment of the angle between the tangents at the ends of the chord. 27. If two rectangular hyperbolas intersect in A, B, C, D ; the circles described on AB, CD as diameters intersect each other ortho- gonally. 28. Prove that the triangle, formed by the tangent at any point and its intercepts on the axes, is similar to the triangle formed by the straight line joining that point with the centre, and the abscissa and ordinate of the point. 29. The angle of inchnation of two tangents to a parabola is half a right angle ; prove that the locus of their point of intersection is a rectangular hyperbola, having one focus and the corresponding directrix coincident with the focus and directrix of the parabola. 30. P is a point on the curve, and PM, PN are straight lines making equal angles with one of the asymptotes ; if MP, NP be pro- duced to meet the curve in P" and Q!, then PQ! passes through the centre. 31. A circle and a rectangular hyperbola intersect in four points and one of their common chords is a diameter of the hyperbola ; shew that the other common chord is a diameter of the circle. 32. AB is & chord of a circle and a diameter of a rectangular hyperbola ; P any point on the circle ; AP, BP, produced if necessary, meet the hyperbola in Q, Q', respectively ; the point of intersection of BQ, AQ! wUl be on the circle. 33. PP is any diameter, Q any point on the curve, PR, PR' are drawn at right angles to PQ, PQ respectively, intersecting the normal at Q in R, R'; prove that QR and QR' are equaL 34. Parallel tangents are drawn to a series of confocal ellipses; prove that the locus of the points of contact is a rectangular hyperbola having one of its asymptotes parallel to the tangents. 35. If tangents, parallel to a given direction, are drawn to a system of circles passing through two fixed points, the points of contact lie on a rectangular hyperbola. EXAMPLES. 135 36. If from a point P on the curve chords are equally inclined to the asymptotes, the line joining their other extremities passes through the centre. 37. From the point of intersection of the directrix with one of the asymptotes of a rectangular hyperbola a tangent is drawn to the curve and meets the other asymptote in T : shew that CT is equal to the transverse axis. 38. The normals at the ends of two conjugate diameters intersect on the asymptote, and are parallel to another pair of conjugate diameters. 39. If the base AB of a triangle ABC be fixed, and if the dififer- ence of the angles at the base is constant, the locus of the vertex is a rectangular hyperbola. 40. A circle described through the angular points A, B oi a, given triangle ABC meets AC in D. If BD meet the tangent at A in P, shew that the vertex and orthocentre of the triangle APB lie on fixed rectangular hyperbolas. 41. The locus of the point of intersection of tangents to an ellipse which make equal angles with the transverse and conjugate axes respectively, and are not at right angles, is a rectangular hyperbola whose vertices are the foci of the ellipse. 42. If OT is the tangent at the point O of a rectangular hyperbola, and PQ a chord meeting it at right angles in T, the two bisectors of the angle OCT bisect OP and OQ. 43. With two sides of a square as asymptotes, and the opposite point as focus, a rectangular hyperbola is described; prove that it bisects the other sides. 44. With the focus S of a. rectangular hyperbola as centre and radius equal to SC a circle is described, prove that it touches the conjugate hyperbola. 45. If parallel normal chords are drawn to a rectangular hyper- bola, the diameter bisecting them is perpendicular to the join of their feet. 46. From the foot of the ordinate PiVof a point P of a rectangular hyperbola, tangents iV§, iViJ are drawn to the circle on A A' as diameter. Prove that PQ passes through A', and PR through A, and that, if QR intersect AA' in M, PM is the tangent at P. 47. 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. 136 EXAMPLES. 48. Through a point P on an equilateral hyperbola two lines are drawn parallel to a pair of conjugate diameters ; the one meeting the curve in P, P", and the other meeting the asymptotes in Q, §'; shew thatPP' = W. 49. If four points forming a parallelogram be taken on a rect- angular hyperbola, then the product of the perpendiculars from any point of the curve on one pair of opposite sides equals the product of the per jjendiculars on the other pair of sides. CHAPTER VI. The Cylinder and the Cone. definition. 145. If a straight line move so as to pass through the circumference of a given circle, and to be perpendicular to the plane of the circle, it traces out a surface called a Right Circular Gylinder. The straight line drawn through the centre of the circle perpendicular to its plane is the Aaris of the Cylinder. It is evident that a section of the surface by a plane perpendicular to the axis is a circle, and that a section by any plane parallel to the axis consists of two parallel lines. Paop. I. Any section of a cylinder by a plane not parallel or perpendicular to the axis is an ellipse. If APA' be the section, let the plane of the paper be the plane through the axis perpendicular to APA'. Inscribe in the cylinder a sphere touching the cylinder in the circle .^i^and the plane APA' in the point S. Let the planes APA', EF intersect in XK, and from any point P of the section draw PK perpendicular to XK. Draw through P the circular section QP, cutting APA' in PN, so that PN is at right angles to A A' and therefore parallel to XK. Let the generating line through P meet the circle EF in R ; and join SP. Then PS and PR are tangents to the sphere ; .•.SP = PR = EQ. 138 But and ^ •.■iSA-.AX, NX=PK, .-.SP'.PKv.SA -.AX. Also, AE being less than AX, 8A is less than AX, and the curve APA' is therefore an ellipse, of which^-S is the focus and XK the directrix. If anotjijer B{)here be inscribed in the cylinder touching A A' in /ST, S' is the other focus, and the corresponding di- rectrix is tl^ intersection of' the plane of contact E'P' with , APA'. ■ , Producing the generating line RP to meet the circle ET in R we pbserve that ^P = PB', and therefore SP~¥S'P = iR=EE' . =AE+AE' = A8 + AS'; and AS' = AE' = A'F= A'S, .■.8P+S'P = AA'. THE CTLIKDMft AKD THE CONE. 139 The transverse axi& of the sectioa is AA!^ an^d the ^on- jugate, or minor, axis is evidently, a diameter of a oscular section. ' . tP ,. "'' 146. Def. If be a fixed point in a straight liijie OE drawn through the centre ^ of a fixed circle 4t right angles to the plane of the circle, and if a stiraight line yOP move so as always to pass through the circurnference of the circle, the surface' generated by the libe (^OP is called a Right Circular Cone. The line OE is called the axis of the cone, the point is the vertex, and the constant angle POE is the semi-vertical angle of the cone. It is evident that any section by a plane perpendicular to the axis, or parallel to the base of the cone, is a circle ; and that any section bya plane through the vecte» consists of two straight lines, the angle between wdfcich is greatest and equal to the vertical angle wheik the plane contains the axis. Any plane containing the:, axis is called a Principal Section. 140 THE GONE. 147. Prop. II. The section of a cone by a plane, which is not perpendicvlar to the axis, and does not pass through the vertex, is either an Ellipse, a Parabola, or an Hyperbola. Let UAP be the cutting plane, and let the plane of the paper be that principal section which is perpendicular to the plane UAP; OV, OAQ being the generating lines in the plane of the paper. Let AUhe the intersection of the principal section VOQ by the plane PA U perpendicular to it, and cutting the cone in the curve AP. Inscribe a sphere in the cone, touching the cone in the circle EF and the plane AP in the point S, and let XK be the intersection of the planes AP, EF. Then XK is per- pendicular to the plane of the paper. Taking any point P in the curve, join OP cutting the circle EF in R, and join SP. Draw through P the circular section QPV cutting the plane AP in PN which is therefore perpendicular to ^JV and parallel to XK. Then, SP and PR being tangents to the sphere, SP = PR = EQ; and Also THE CONE. EQ : i\^X :: AE : AX :: AS : AX. NX = PK; :. SP : PK :: SA : AX. 141 The curve AP is therefore an Ellipse, Parabola, or Hy- perbola, according as SA is less than, equal to, or greater than AX. In any case the point iS is a. focus and the cor- responding directrix is the intersection of the plane of the curve with the plane of contact of the sphere. 148. (1) li AUhe parallel to OV, the angle AXE = OFE= OEF=AEX, so that SA = AE=AX ; the section is therefore a parabola when the cutting plane is parallel to a generating line, and perpendicular to the principal section which contains the generating line. (2) Let the line A U meet the curve again in the point A' on the same side of the vertex as the point A. 142 THE CONE. Then the angle AEX^OFX ^FXA, and therefore AE AX, that is AS>AX, and the curve AP is one branch of an hyperbola, the other branch being the section A'P". Taking P" in the other branch the proof is the same as before that SF : FK' :: 8A : AX. In this case a sphere can be inscribed in the other branch of the cone, touching the cone along the circle E'F', and the plane UA'P' in S', and it can be shewn that 8' is the other focus of the hyperbola, and that the directrix is the intersection of the cutting plane with the plane of con- tact E'F. Hence the section of a cone by a plane cutting in AU THE CONE. 143 the principal section VOQ perpendicular to it is an Ellipse, Parabola, or Hyperbola, according as the angle EAX is greater than, equal to, or less than, the vertical angle of the cone. Further, it is obvious that, if any plane be drawn parallel to the plane AP, the ratio oiAE to AX is always the same; hence it follows that all parallel sections have the same eccentricity. 149. This method of determining the focus and directrix was published by Mr Pierce Morton, of Trinity College, in the first volume of the Cambridge Philosophical Transactions. The method was very nearly obtained by Hamilton, who gave the following construction. First finding the vertex and focus, A and S, take AE along the generating line equal to AS, and draw the circular 144 THE CONE. section through E; the directrix will be the line of inter- section of the plane of the circle with the given plane of section. Hamilton also demonstrated the equality of SP and PR. 150. Prop. III. To prove that, in the case of an elliptic section, SP + S'P^AA'. Taking the 2nd figure, SP = PR and 8'P = PE'; .■.8P + 8'P = RR'='EE = AE-¥AE' = AS+AS'. But A'S' = A'F = FF-A'F = EE-A'8, also A'S' + 8S'=A'S; .■.2A'S' + 8S' = EE'. Similarly 2A8 + 88' = EE; .•.A'S' = A8, and A^ = A'8. Hence 8P+S'P = AA'. and the transverse, or major axi8 = ^^'. In a similar manner it can be shewn that in an hyper- bolic section S'P-SP = AA'. 1.51. Prop. IV. To shew that, in a paraholic section, PN' = iAS.AN. Let A be the vertex of the section, and let ADE be the diameter of the circular section through A. THE CONK From D let fall BS perpendicular to AN then PN' = QN.NQf = QN.AE = 4>NL.AD, if AL be perpendicular to NQ. But the triangles ANL, ADS being similar, NL:AK::AS:AD; .-.NL.AD^AN .AS, 145 and PN' = 4A.S.AK 152. Prop. V. To shew that, in an elliptic section, PN' is to AN . NA' in a constant ratio. Draw through P the circular section QPQ', bisect AA' in G, and draw through G the circu- lar section EBE'. Then QN : AN :: GE : AG. and NQ':NA'::GE':A'G; .-.QN.Nq -.AN.NA' EG . GE' : AG\ or PN':AN.NA'::EG.CE:AC^; and, the transverse axis being AA', the square of the semi- minor axis = BC^ = EG . GE'. ■RPR 10 146 THE CONE. Again, if ADF be perpendicular to the axis, AD = DF, and, AG being equal to vA', CD is parallel to AF, and therefore CE' = FD = AD. Similarly, CE=A'D', the perpendicular from A' on the axis ; .■.BC' = AD.A'D', that is, the semi-minor axis is a mean proportional between the perpendiculars from the vertices on the axis of the cone. Cor. If H, H' are the centres of the focal spheres, the angles HAH", HA'H' are right angles, so that H, A, H', A' are concyclic. It follows that the triangles ASH, A'H'D' are similar, as are also the triangles A'S'H', AHD, so that SH : A'D' :: AH : A'H' :: AD : S'H' ; and SH . S'H' = AD . A'D' = BC ; .". the semi-minor axis is a mean proportional between the radii of the focal spheres. THE CONE. 147 The fact that H, A, W, A' are concyclic also shews that the sphere of which HH' is a diameter intersects the plane of the ellipse in its auxiliary circle. 153. In exactly the same manner it can be shewn that, for an hyperbolic section, and that PJV' : AN . NA' :-. CE . CK : AC, CE=AD, and CE = A'U. Also, as in the case of the ellipse, BG is a mean pro- portional between AD and A'ly, and is also a mean pro- portional between the radii of the focal spheres. 154. Pkop. VI. The two straight lines in which a cone is intersected by a plane through the vertex parallel to an hyperbolic section are parallel to the asymptotes of the hyperbola. Taking the preceding figure, let the parallel plane cut 10—2 148 THE CONE. the cone in the lines OG, OG', and the circular section through C in the line GL6', which will be perpendicular to the plane of the paper, and therefore perpendicular to US' and to OL. Hence GU = EL.E 'L. But EL : EC :: OL ■.A'C. and E'L : EC :: OL :AG; :. GL' :EC .JEC:: OL' ■.AC, or GL : OL :: BC ■.AC; therefore, (Art. 102),0(? and OG'are parallel to the asymptotes of the hyperbola. Hence, for all parallel hyperbolic sections, the asymptotes are parallel to each other. If the hyperbola be rectangular, the angle GOG' is a right angle ; but this is evidently not possible if the vertical angle of the cone be less than a right angle. When the vertical angle of the cone is not less than a right angle, and when GOG' is a right angle, LOG is half a right angle, and therefore OL = LG, and 1.0U = 0G^=0E', and the length OL is easily constructed. Hence, placing OL, and drawing the plane GOG' per- pendicular to the principal section through OL, any section by a plane parallel to GOG' is a rectangular hyperbola. It will be observed that the eccentricity of the section is greatest when its plane is parallel to the axis of the cone. 155. Prop. VII. The sphere which passes throtigh the circles of contact of the focal spheres with the surface of the cone intersects the plane of the section in its director circle. Let Q, Q be the points in which the straight line AA' is intersected by the sphere which passes through the circles EF and EF. THE CONE. 149 Then the sphere intersects the plane of the ellipse in the circle of which QQf is the diameter. Also CQ'- CA* = AQ.Aq = AE.AE' = AS.AS' = BG^; :.CQ' = AC* + BC\ so that CQ is the radius of the director circle. Changing the figure the proof is exactly the same for the typerbola. 156. Prop. VIII. If two straight lines be drawn through any point, parallel to two fixed lines, aiid intersecting a given cone, the ratio of the rectangles formed by the segments of the lines will be independent of the position of tlie point. Thus, if through E, the lines EPQ, EFQ' be drawn, parallel to two given lines, and cutting the cone in the 150 THE CONE. points P, Q and P', Q', the ratio of EP.EQ to EF.EQf is constant. Through draw OK parallel to the given line to which EPQ is parallel, and let the plane through OK, EPQ, which contains the generating lines OP, OQ, meet the circular section through E in R and S, and the plane base in the straight Une DFK, cutting the circular base in D and P. Then DFK and ERS being sections of parallel planes by a plane are parallel to each other. Also, EPQ is parallel to OK; Therefore ERP, ODK are similar triangles, as are also ESQ, OFK; :. EP : ER :: OK : DK, and EQ:E8::0K:FK; .: EP . EQ : ER . ES :: OK' : DK . FK ::0K': KT', if KT be the tangent to the circular base from K. If a similar construction be made for EP'Qf, we shall have EP' . EQf : ER' . ES' :: OK'^ : K'T''. THE CONB. 151 But ER.ES = ER' .ES'; therefore the rectangles EP . EQ and EP" . EQ^ are each in a constant ratio to the same rectangle, and are therefore in a constant ratio to each other. Since the plane through EPQ, EP'Q' cuts the cone in an ellipse, parabola, or hyperbola, this theorem includes as particular cases those of Arts. 51, 58, 82, 92, 96, 124 and 134. The proof is the same if the point P be within the cone, or if one or both of the lines meet opposite branches of the cone. If the chords be drawn through the centre of the section PEP', the rectangles become the squares of the semi-diameters. Hence the parallel diameters of all parallel sections of a cone are proportional to each other. If the lines move until they become tangents the rect- angles then become the squares of the tangents ; therefore if a series of points be so taken that the tangents from them are parallel to given lines, these tangents are always in the same proportion. The locus of the point E will be the line of intersection of two fixed planes touching the cone, that is, a fixed line through the vertex. EXAMPLES. 1. Shew how to cut from a cylinder an ellipse of given eccentricity. 2. What is the locus of the foci of all sections of a cylinder of a given eccentricity ? 3. Shew how to cut from a cone an elUpse of given eccentricity. 4. Prove that all sections of a cone by parallel planes are conies of the same eccentricity. 5. What is the locus of the foci of the sections made by planes inclined to the axis at the same angle i 152 EXAMPLES. 6. Find the least angle of a cone from \rhich it is possible to cut an hyperbola, whose eccentricity shall be the ratio of two to one. 7. The centre of a spherical ball is moveable in a vertical plane which is equidistant from two candles of the same height on a table ; find its locus when the two shadows on the ceiling are always just in contact. 8. Through a given point draw a plane cutting a given cone in a section which has the given point for a focus. 9. If the vertical angle of a cone, vertex 0, be a right angle, P any point of a parabolic section, and PN perpendicular to the aiis of the parabola, OP='iAS+AJ^, A being the vertex and S the focus. 10. Prove that the directrices of all parabolic sections of a cone lie in the tangent planes of a cone having the same axis. 11. If the curve formed by the intersection of any plane with a cone be projected upon a plane perpendicular to the axis ; prove that the curve of projection will be a conic section having its focus at the point in which the axis meets the plane of projection. 12. Prove that the latera recta of parabolic sections of a right circular cone are proportional to the distances of their vertices from the vertex of the cone. 13. The shadow of a ball is cast by a candle on an inclined plane in contact with the ball ; prove that as the candle bums down, the locus of the centre of the shadow will be a straight line. 14. The vertex of a right cone which contains a given ellipse lies on a certain hyx>erbola, and the axis of the cone will be a tangent to the hyperbola. 15. Find the locus of the vertices of the right circular cones which can be drawn so as to peiss through a given fixed hyperbola, and prove that the axis of the cone is always tangential to the locus. 16. An ellipse and an hyperbola are so situated that the vertices of each ciirve are the foci of the other, and the curves are in planes at right angles to each other. K P be a point on the ellipse, and a point on the hyperbola, S the vertex, and A the interior focus of that branch of the hyperbola, then AS^OP = AO + SP. 17. The latus rectum of any plane section of a given pone is pro- portional to the perpendicular from the vertex on the plane. 18. If a sphere is described about the vertex of a right cone as centre, the latera recta of aU sections made by tangent planes to the sphere are equal. EXAMPLES. 153 19. Different elliptic sections of a right cone are taken such that their minor axes are equal ; shew that the locus of their centres is the surface formed by the revolution of an hyperbola about the axis of the cone. 20. If two cones be described touching the same two spheres, the , eccentricities of the two sections of them made by the same plane bear to one another a ratio constant for all positions of the plane. 21. If elliptic sections of a cone be made such that the volume between the vertex and the section is always the same, the minor axis will be always of the same length. 22. The vertex of a cone and the centre of a sphere inscribed within it are given in position : a plane section of the cone, at right angles to any generating line of the cone, touches the sphere : prove that the locus of the point of contact is a surface generated by the revolution of a circle, which touches the axis of the cone at the centre of the sphere. 23. Given a right cone and a point within it, there are two sections which have this point for focus ; and the planes of these sections make equal angles with the straight line joining the given point and the vertex of the cone. 24. Prove that the centres of all plane sections of a cone, for which the distance between the foci is the same, lie on the surface of a right circular cylinder. CHAPTER VII. The Similarity of Conics, the Areas of Conics, and THE Curvatures of Conics. SIMILAR CONICS. 157. Def. Conics which have the same eccentricity are said to be similar to each other. This definition is justified by the consideration that the character of the conic depends on its eccentricity alone, while the dimensions of all parts of the conic are entirely determined by the distance of the focus from the directrix. Hence, according to this definition, all parabolas are similar curves. Prop. I. If radii be drawn from the vertices of two parabolas making equal angles with the axis, these radii are always in the same proportion. Let AP, ap be the radii, PN and pn the ordinates, the angles PAN, pan, being equal. Then AP* : ap' :: PN* : pn" :: AS . AN : as . an. But AP : ap :: AN : an ; .'. AP : ap :: AS : as. It can also be shewn that focal radii making equal angles with the axes are always in the same proportion. SIMILAR CONICS. 155 158. Prop. II. If two ellipses be similar their axes are in the same proportion, and any other diameters, making equal angles with the respective axes, are in the proportion of the axes. Let CA, GB be the semi-axes of one ellipse, ca, cb of the other, and GP, cp two radii such that the angle PGA =pca. Then, since the eccentricities are the same, we have, if S, s be foci, AG : 8G :: ac : sc; .: AG" : AG'-SG" :: ad> : ad'-sc\ or AG' : BG^ :: ad' : be'. Hence it follows, if PN, pn be ordinates, that PF' : AG' - GJSr ■.■.pn' : ac' - en' ; but, by similar triangles, PK : pn :: GN : en, therefore CN' : AG'-GN' :: en' : ac' - en' ; and GJT : AG' :: en' : ac'. Hence GP : cp :: GN : en :: AG : ac. So also lines drawn similarly from the foci, or any other corresponding points of the two figures, will be in the ratio of the transverse axes. Exactly the same demonstration is applicable to the hyperbola, but in this case, if the ratio of 8G to AG in two hyperbolas be the same, it follows from Art. (102) that the angle between the asymptotes is the same in both curves. In the case of hyperbolas we have thus a very simple test of similarity. 156 AREAS. The Areas bounded hy Conies. 159. Prop. III. 1/ AB, AG be two tangents to a para- bola, the area between the curve and the chord BC is two-thirds of the triangle ABC. Draw the tangent DPU parallel to BG; then AP = PN, and BG=2.I)E; therefore the triangle BPG= 2ADE. Again, draw the diameter DQM meeting BP in M. By the same reasoning, FQO being the tangent parallel to BP, the triangle PQB = 2FDQ. Through F draw the diameter FRL, meeting PQ in L, and let this process be continued indefinitely. Then the sum of the triangles within the parabola is double the sum of the triangles without it. AREAS. 157 But, since the triangle BPG is half ABC, it is greater than half the parabolic area BQPG ; Therefore (Euclid, Bk. xii.) the difference between the parabolic area and the sum of the triangles can be made ultimately less than any assignable quantity ; And, the same being true of the outer triangles, it follows that the area between the curve and BO is double of the area between the curve and AB, AG, and is therefore two- thirds of the triangle ABG. CoK. Since PN bisects every chord parallel to BG, it bisects the parabolic area BPG; therefore, completing the parallelogram PNBU, the parabolic area BPN ia two-thirds of the parallelogram UN. 160. Pkop. IV. The area of an ellipse is to the area of the atuciliary circle in the ratio of the conjugate to the trans- verse axis. Draw a series of ordinates, QPN, QP'N',... near each other, and draw PR, QR' parallel to AG. Then, since PN : QN :: BG : AG. the area PN' : QN' :: BG : AG, and, this being true for all such areas, the sum of the parallelograms PN' is to the sum of the parallelograms QN' as BG to AG. But, if the number be increased indefinitely, the sums of 158 ASEAS. these parallelograms ultimately approximate to the areas of the ellipse and circle. Hence the ellipse is to the circle in the ratio of BC to AG. .\> ' The student will find in Newton's 2nd and 3rd Lemmas ■ ■ (Prindpia, Section i.) a formal proof of what we have here assumed as sufiBcientlj obvious, that the sum of the paral- lelograms PN is ultimately equal to the area of the ellipse. 161. Pkop. Y. If P, Q be two points of an hyperbola, and if PL, QM parallel to one asymptote meet the other in L and M, the hyperbolic sector CPQ is equal to the hyperbolic trapezium PLMQ. For the triangles GPL, GQM are equal, and, if PL meet CQ in R, it follows that the triangle GPR = the trapezium LRQM; hence, adding to each the area RPQ, the theorem is proved. 162. Pkop. YI. If points L, M, N, K be taken in an asymptote of an hyperbola, such that GL : CM :: CN : CK, and if LP, MQ, NR, KS, parallel to the asymptote, meet the curve in P, Q, R, S, the hyperbolic areas CPQ, CR8 will be Let QR and PS produced meet the asymptotes in F, F', G.Q'; ABEAS. 159 then RF= QF and SG = PG' (Art. 121), .-. FF= CM and KG = CL. Hence IfF:KG::CM:CL GK :GN RN-.SK, and therefore SP is parallel to QR. %i .PT. Draw the ordinate RNP"; then taking the figure of the next article, TP' is the tangent at P", and the angle FTF= PTF== PFT ; therefore PQ is parallel to TP", and is bisected by the diameter FE. Hence PQ = 2 . PE = 4P'T = 4SP : PL iSP . PL : PL' ; but iSP.PL = 8SP.AS=887* = 2PG'; .: PO : PG :: 2P(?' : PL". 166. Prop. X. If the chord of intersection, PQ, of an ellipse, or hyperbola, with the circle of curvature at P, meet CD, the semidiameter conjugate to CP, in K, PQ.PK=2CD'. Drawing the ordinate PNP', the tangent at P' is parallel to PQ, as in the parabola, and PQ is therefore bisected in V, by the diameter CP". Let PQ meet the axes in U and U'; then, U'G being parallel to PF, CURVATURE. PV:PU' :: VF : GF :: UT : CT, since PTJ, P'T are parallel. Also UT: GT::PU:PK; .: PV : PU' :: PU : PK. Hence PV.PK = PU.PU' = PT.PT' = CB', observing that PU = PT, and T' PU' = PT, by the theorem of Art. 163, and :.PQ.PK=2CD\ 163 167. Prop. XI. If the chord of curvature PQ', of an ellipse or hyperbola in any direction, meet CD in K', Pi^ .PK' = 2CJy. Let PO be the diameter of curvature meeting CB in F: 11—2 164 CURVATURE. then PQO, PQO are right angles, and a circle can be drawn through QK'FO; .■.PQ'.PK' = PF .PO = PK.PQ = 2.CD\ Cor. 1. Hence PO being the diameter of curvature, PF.P0 = 2. CD'. Cor. 2. If PQ' pass through the focus, PK' = AG, and PQ'.ylC=2. CD^ Cor. 3. li PQf pass through the centre, PQ .GP = 2.GPP. 168. We can also express the diameter of curvature as follows : PG being the normal, let GL be perpendicular to SP, and let PR be the chord of curvature through S. Then GL is parallel to OR, and PO : PG :: PR : PL :: PR. PL: PL". But PR. AC =2. CD"; .: PR: AC:: 2. CD' : AC ::2.PG^:BC\ and PR.PL:AC.PL::2.PG':BG\ But, PL being equal to the semi-latus rectum, PL.AC = BC'; :.PR.PL = 2.PG\ and PO : PG :: 2PG' : PL\ Hence, in any conic, the radius of curvature at any point is to the normal at the point as the square of the normal to the square of the semi-latus rectum. EXAMPLES. 165 169. Prop. XII. The chord of curvature through the focus at any point is equal to the focal chord parallel to the tangent at the point. Since P(^ .AG=2CI)\ it follows that PQ' .AA' = DD'\ But, iipp' is the focal chord parallel to the tangent at P, ^'.^^' = DD"(Art. 81), .■.PQ=PP'. EXAMPLES. 1. The radius of curvature at the end of the latus rectum of a parabola is equal to twice the normal. 2. The circle of curvature at the end of the latus rectum intersects the parabola on the normal at that point. 3. If P Vis the chord of curvature through the focus, what is the locus of the point V ? 4. An eUipse and a parabola, whose axes are parallel, have the same curvature at a point P and cut one another in ^ ; if the tangent at P meets the axis of the parabola in T prove that PQ=i . PT. 5. In a rectangular hyperbola, the radius of curvature at P varies asCi". 6. If P be a point of an ellipse equidistant from the axis minor and one of the directrices, the circle of ciuT^ature at P will pass through caie of the foci. 7. If the normal at a point P of a parabola meet the directrix in L, the radius of curvature at P is equal to 2 . PL. 8. The normal at any point P of a rectangular hyperbola meets the curve again in Q ; shew that PQ is equal to the diameter of curva- ture at P. 9. In the rectangular hyperbola, if CP be produced to Q, so that PQ= CP, and QO be drawn perpendicular to CQ to intersect the normal in 0, is the centre of curvature at P. 166 EXAMPLES. 10. At any point of aa ellipse the chord of curvature PV through the centre is to the focal chord 'pp\ parallel to the tangent, as the major axis is to the diameter through the point. 11. If the common tangent of an ellipse and its circle of curvature at P be bisected by their common chord, prove that Cm=AC.BC. 12. The tangent at a point P of an ellipse whose centre is C meets the axes in T and t ; if CP produced meet in L the circle described about the triangle TCt, shew that PL is half the chord of curvatiu-e at P in the direction of C, and that the rectangle contained by CP, CL, is constant. 13. If f be a point on a conic, Q a point near it, and if QE, per- pendicular to PQ, meet the normal at P in E, then ultimately when Q coincides with P, PE is the diameter of curvature at P. 14. If a tangent be drawn from any point of a parabola to the circle of curvature at the vertex, the length of the tangent will be equal to the abscissa of the point measured along the axis. 15. The circle of curvature at a point where the conjugate diameters are equal, meets the ellipse again at the extremity of the diameter. 16. The chord of curvature at P perpendicular to the major axis is to PM, the ordinate at /», : : 2 . CD^ : BC IT. Prove that there is a point P on an ellipse such that if the normal at P meet the ellipse in Q, PQ is a chord of the circle of curva- ture at P, and find its position. 18. The chord of curvature at a point /" of a rectangular hyper- bola, perpendicular to an asymptote, is to CD : : CD : 2 . PJV^, where PiV is the distance of P from the asymptote. 19. If & be the foot of the normal at a point P of an ellipse, and GK, perpendicular to P6, meet CP in E, then KE, parallel to the axis minor, will meet PG in the centre of curvature at P. 20. The chord of curvature through the vertex at a point of a para- bola is to iPY :: PF : AP. 21. Prove that the locus of the middle points of the common chords of a given parabola and its circles of curvature is a parabola, and that the envelope of the chords is also a parabola. 22. The circles of curvature at the extremities P, D of two con- jugate diameters of an ellipse meet the ellipse again in Q, R, respec- tively, shew that PB, is parallel to DQ. 23. The tangent at any point P in an ellipse, of which S and H are the foci, meets the axis major in T, and TQR bisects HP in Q and EXAAtPLES. 167 meets SP in It ; prove that PR is one-fourth of the chord of curvature at P through S. 24. An ellipse, a parabola, and an hyperbola, have the same vertex and the same focus ; shew that the curvature, at the vertei, of the parabola is greater than that of the hyperbola, and less than that of the ellipse. 25. The circle of curvature at a point of an ellipse cuts the curve in Q ; the tangent at P is met by the other common tangent, which touches the curves at E and F,inT; if PQ meet TEF in 0, TEOF is cut harmonically. 26. If E is the centre of curvature at the point P of a parabola, SE^ + S . SP^=^PEK 27. Find the locus of the foci of the parabolas which have a given circle as circle of curvature, at a given point of that circle. 28. Two parabolas, whose latera recta have a constant ratio, and whose foci are two given points A, B, have a contact of the second order at P. Shew that the locus of i* is a circle. 29. If the fixed straight line PQ is the chord of an ellipse, and is also the diameter of curvature at P, prove that the locus of the centre of the ellipse is a rectangular hyperbola, the transverse axis of which is coincident in direction with PQ, and equal in length to one- half of P^. CHAPTER VIII. Orthogonal Projections. 170. Def. The projection of a point on a plane is the foot of the perpendicular let fall from the point on the plane. If from all points of a given curve perpendiculars be let fall on a plane, the curve formed by the feet of the perpen- diculars is the projection of the given curve. The projection of a straight line is also a straight line, for it is the line of intersection with the given plane of a plane through the line perpendicular to the given plane. Parallel straight lines project into parallel lines, for the projections are the lines of intersection of parallel planes with the given plane. 171. Prop. I. Parallel straight lines, of finite lengths, are projected in the same ratio. That is, if ab, pq be the projections of the parallel lines AB,FQ. ab : AB :: pq : PQ. For, drawing AC parallel to ab and meeting Bb in C, and PR parallel to pq and meeting Qq in .B, .4J5Cf and PQR are similar triangles ; therefore AC : AB :: PR : PQ, and AG = ab, PR = pq. 172. Prop. II. The projection of the tangent to a curve at any point is the tangent to the projection of the curve at the projection of the point. PROJECTIONS. 169 For if p, q be the projections of the two points P, Q of a curve, the line pq is the projection of the line PQ, and when the line PQ turns round P until Q coincides with P, pq turns round p until q coincides with p, and the ultimate position oipq is the tangent a.tp. 173. Prop. III. The projection of a circle is an ellipse. Let aba' be the projection of a circle ABA '. Take a chord PQ parallel to the plane of projection, then its projection pq = PQ. Let the diameter ANA' perpendicular to PQ meet in F the plane of projection, and let aa'F be the projection of AA'F. Then aa' bisects pq at right angles in the point n, and an : AN :: aF : AF, a'n : A'N :: aF : AF; .: AN . NA' : an . na' :: AF" : aF'; but AN . NA' = FN' = pn\ :. pr^ : an . na' :: AF* : aF*, and the curve apa' is an ellipse, having its axes in the ratio of aF : AF,oro{aa' : AA'. 170 PKOJECTIONS. Moreover, since we can place the circle so as to make the ratio of oua^ to AA' whatever we please, an ellipse of any eccentricity can be obtained. In this demonstration we have assumed only the pro- perty of the principal diameters of an ellipse. Properties of other diameters can be obtained by help of the preceding theorems, as in the following instances. 174. Pkop. IV. The locus of the middle points of parallel chords of an ellipse is a straight line. For, projecting a circle, the parallel chords of the ellipse are the projections of parallel chords of the circle, and as the middle points of these latter lie in a diameter of the circle, the middle points of the chords of the ellipse lie in the pro- jection of the diameter, which is a straight line, and is a diameter of the ellipse. Moreover, the diameter of the circle is perpendicular to the chords it bisects ; hence Perpendicular diameters of a circle prefect into conjugate diameters of an ellipse. 175. Prop. V. If two intersecting chords of an ellipse he parallel to fixed lines, the ratio of the rectangles contained hy their segments is constant. Let OPQ, ORS be two chords of a circle, parallel to fixed lines, and opq, ors their projections. Then OP . OQ is to op . oq in a constant ratio, and OR . OS is to or .OS in & constant ratio ; but OP.OQ = OR.OS. Therefore op . oq is to or .os in a constant ratio ; and opq, ors are parallel to fixed lines. 176. Prop. VI. If qvq' be a double ordinate of a diameter cp, and if the tangent at q meet cp produced in t, cv.ct = cp*. The lines qvq^ and cp are the projections of a chord QVQ of a circle which is bisected by a diameter CP, and t is the PROJECTIONS. 171 projection of T the point in which the tangent at Q meets CP produced. But, in the circle, CV . GT=Cr, or GV : CP :: GP : GT; and, these lines being projected in the same ratio, it follows that CV : cp :: cp : ct, or CV . ct = cp'. Hence it follows that tangents to an ellipse at the ends of any chord meet in the diameter conjugate to the chord. The preceding articles will shew the utility of the method in dealing with many of the properties of an ellipse. The student will find it useful to prove, by orthogonal projections, the theorems of Arts. 58, 69, 74, 75, 78, 79, 80, 82, 83, 89, 90, and 92. 177. Prop. VII. An ellipse can be projected into a circle. This is really the converse of Art. 173, but we give a construction for the purpose. Draw a plane through AA', the transverse axis, perpendicular to the plane of the ellipse, and in this plane describe a circle on AA' as diameter. Also take the chord AB, equal to the conjugate axis, and join A'D, which is per- pendicular to AB. Through AD draw a plane perpendicular to A'B, and pro- ject a principal chord PNP' on this plane. Then PN' : AN . NA' :: BG' : AG\ I \. ^^ A V P. p But PN=pn, PROJECTIONS. An : AN :: AD : AA' :: BC AC, Dn : A'N :: BC AC. nD : AN . NA' :: BC 172 and Hence An . nD : AN . NA' :: BC : AC, and therefore pi' = An . nD, and the projection ApD is a circle. This theorem, in the same manner as that of Art. 173, may be employed in deducing properties of oblique diameters and oblique chords of an ellipse. 178. If any figures in one plane be projected on another plane, the areas of the projections will all he in the same ratio to the areas of ilte figures themselves. Let BAD be the plane of the figures, and let them be projected on the plane CAD, C being the projection of the point B, and BAD being a right angle. Taking a rectangle EFOH, the sides of which are parallel and perpendicular to AD, the projection is efgh, and it is clear that the ratio of the areas of these rectangles is that oiACioAB. Now the area of any curvilinear figure in the plane BAD is the sum of the areas of parallelograms such as EFGH, which are inscribed in the figure, if we take the widths, such as EF, infinitesimally small. PROJECTIONS. 173 It follows that the area of the projection of the figure is to the area of the figure itself in the ratio oi AC to AB. As an illustration, let a square be drawn circumscribing a circle, and project the figure on any plane. The square projects into tangents parallel to conjugate diameters of the ellipse which is the projection of the circle. The area of the parallelogram thus formed is the same whatever be the position of the square, and we thus obtain the theorem of Art. 87. 179. It follows that maxima and minima areas project into maxima and minima areas. For example, the greatest triangle which can be inscribed in a circle is an equilateral triangle. Projecting this figure we find that the triangle of maxi- mum area inscribed in an ellipse is such that the tangent at each angular point is parallel to the opposite side, and that the centre of the ellipse is the point of intersection of the lines joining the vertices of the triangle with the middle points of the opposite sides. 180. Peop. VII. The prcfjection of a parabola is a parabola. For if PNP' be a principal chord, bisected by the axis AN, the projection pnp' will be bisected by the projection an. Moreover pn : PN will be a constant ratio, as also will be an : AN. And PN* = 4>AS . AN. Hence pn' will be to 4>AS . an in a constant ratio, and the projection is a parabola, the tangent at a being parallel to pn. 181. Prop. VIII. An hyperbola can be always projected into a rectangular hyperbola. For the asymptotes can be projected into two straight lines cl, cV at right angles, and if PM, PN be parallels to the asymptotes from a point P of the curve, PM . PN is constant. 174 EXAMPLES. But pm : PM and pn : PN are constant ratios ; .". pm . pn is constant. And since pm and pn are perpendicular respectively to cl and cl', it follows that the projection is a rectangular hyperbola. The same proof evidently shews that any projection of an hyperbola is also an hyperbola. EXAMPLES. 1. A parallelogTam is inscribed in a given ellipse; shew that its sides are parallel to conjugate diameters, and find its greatest area. 2. TP, TQ are tangents to an ellipse, and CP', CQ' are parallel semidiameters ; PQ is parallel to P'Q'. 3. If a straight line meet two concentrie similar and similarly situated ellipses, the portions intercepted between the curves are equal. 4. Find the locus of the point of intersection of the tangents at the extremities of pairs of conjugate diameters of an ellipse. 5. Find the locus of the middle points of the lines joining the extremities of conjugate diameters. 6. If a tangent be drawn at the extremity of the major axis meeting two equal conjugate diameters CP, CD produced in T and t; then PJP=2AT^. 7. If a chord AQ drawn from the vertex be produced to meet the minor axis in 0, and CP be a semidiameter parallel to it, then AQ . A0=2CP^. 8. OQ, 0^ are tangents to an ellipse from an external point O, and OR is a diagonal of the parallelogram of which 0§, OQ are adjacent sides ; prove tluit if £ be on the ellipse, will lie on a similar and similarly situated concentric ellipse. 9. ABi'&i. given chord of an ellipse, and C any point in the ellipse ; shew that the locus of the point of intersection of lines drawn from J, 5, C to the middle points of the opposite sides of the triangle ABC is a similar ellipse. 10. CP, CD are conjugate semidiameters of an ellipse ; if an ellipse, similar and similarly situated to the given ellipse, be described on PD as diameter, it will pass through the centre of the given ellipse. EXAMPLES. 175 11. Parallelograms are inscribed in an ellipse and one pair of opposite sides constantly touch a similar, similarly situated and con- centric ellipse ; shew that the remaining pair of sides are tangents to a third ellipse and the square on a principal semi-axis of the original ellipse is equal to the sum of the squares on the corresponding semi- axes of the other two ellipses. 12. Find the locus of the middle point of a chord of an ellipse which cuts off a constant area from the curve. 13. Find the locus of the middle point of a chord of a parabola which cuts off a constant area from the curve. 14. A parallelogram circumscribes an ellipse, touching the curve at the extremities of conjugate diameters, and another parallelogram is formed by joining the points where its diagonals meet the ellipse : prove that the area of the inner parallelogram is half that of the outer one. If four similar and similarly situated ellipses be inscribed in the spaces between the outer parallelogram and the curve, prove that their centres lie in a similar and similarly situated ellipse. 15. About a given triangle PQR is circumscribed an elUpse, having for centre the point of intersection (C) of the lines from P, Q, R bisecting the opposite sides, and PC, QC, RC are produced to meet the curve in i", §*, R'; shew that, if tangents be drawn at these points, the triangle so formed will be similar to PQR, and four times as great. 16. The locus of the middle points of all chords of an ellipse which pass through a fixed point in an ellipse similar and similarly situated to the given ellipse, and with its centre in the middle point of the line joining the given point and the centre of the given ellipse. 17. PT, pt are tangents at the extremities of any diameter Pp of an ellipse; any other diameter meets PTin 2^ and its conjugate meets pt in t; also any tangent meets Py in T' and pt iu t' ; shew that PT :PT' ■.-.pf -.pt. 18. From the ends P, B of conjugate diameters of an ellipse lines are drawn parallel to any tangent line ; from the centre C any line is drawn cutting these lines and the tangent in p, d, t, respectively ; prove tha.t Cp^+C(P=Ct^ 19. If CP, CD be conjugate diameters of an ellipse, and if BP, BD be joined, and also AD, A'P, these latter intersecting in 0, the figure BDOP will be a paraUelogram. 20. 7 is a point on the tangent at a point P of an elUpse, so that a perpendicular from T on the focal distance SP is of constant length ; shew that the locus of 2* is a similar, similarly situated and concentric ellipse. 176 EXAMPLES. 21. Q is a point in one asymptote, and q in the other. If Qq move parallel to itself, find the locus of intersection of tangents to the hyperbola from Q and q. 22. Tangents are drawn to an ellipse from an external point T. The chord of contact and the major axis, or these produced, intersect in K, and T'JV is drawn perpendicular to the major axis. Prove that CJf. CK=CA\ 23. Q is a variable point on the tangent at a fixed point F of an ellipse and R is taken so that PQ=QR. If the other tangent from Q meet the ellipse in K, prove that RK passes through a fixed point. 24. If through any point on an ellipse there be drawn lines con- jugate to the sides of an inscribed triangle they will meet the sides in three points in. a, straight line. 25. PCP" is a diameter of an ellipse, and a chord PQ meets the tangent at i* in R. Prove that FQ, FR have the parallel diameter for a mean proportional. 26. If AOA', BOE are conjugate diameters of an ellipse, and if AP and BQ are parallel chords, A!Q and BF are parallel to conjugate diameters. 27. If the tangents at the ends of a chord of an hyperbola meet in T, and TM, TM' te drawn parallel to the asymptotes to meet them in M, M", then MM' is parallel to the chord. 28. If a windmill in a level field is working uniformly on a sunny day, the speed of the end of the shadow of one sail varies as the length of the shadow of the next sail; 29. Spheres are drawn passing through a fixed point and touching two fixed planes. Prove that the points of contact lie on two circles, and that the locus of the centre of the sphere is an ellipse. If the angle between the planes is the angle of an equilateral triangle, prove that the distance between the foci of the ellipse is half the major axis. CHAPTER IX. Of Conics in Genekal. The Construction of a Conic. 182. The method of construction, given in Chapter I., can be extended in the following manner. Let ySn be any straight line drawn through the focus S, and draw Ax from the vertex parallel to j^, and meeting the directrix in x. Divide the line^n in a and a' so that Sa : af :: Sa' : a'f :: SA : Ax; then a and a are points on the curve, for, if ak be the perpendicular on the directrix, ak : af :: AX : Ax, and therefore Sa : ak :: SA : AX. 12 178 OF CONICS IN GENERAL. Take any point e in the directrix, draw the lines eSl, ea through S and a, and draw SP making the angle PSl equal to ISn. Through P draw FPl parallel to fS, and meeting eS produced in I, then Pl = SP, and PI : PF :: Sa : af; :. SP : PF :: Sa : af, and SP ■ PK :: Sa : ak; therefore P is a point in the curve. 183. The construction for the point a gives a simple proof that the tangent at the vertex is perpendicular to the axis. For when the angle ASa is diminished, Sa approaches to equality with SA, and therefore the angle a AS is ulti- mately a right angle. 184. Prop. I. To find the points in which a given straight line is intersected by a conic of which the focus, the directrix, and the eccentricity are given. OF CONICS IN GENEKAl. 179 Let FPP' be the straight line, and draw Aw parallel to it Join FS, and find the points D and E such that SD : DF :: SE : EF :: SA :. Ax. Describe the circle on DE as diameter, and let it inter- sect the given line in P and P*. Join DP, EP and draw SG, FH at right angles to EP. Then DPE, being the angle in a semicircle, is a right angle, and DP is parallel to SG and FH. Hence SG : FH SE : EF SD : DF PG : PH; therefore the angles SPG, FPH are equal, and therefore PD bisects the angle SPF. Hence SP : PF :: SD : DF :: SA : Ax, and P is a point in the curve. Similarly P" is also a point in the curve, and the per- pendicular from 0, the centre of the circle, on FPP' meets it in V, the middle point of the chord PP. Since SE : EF :: SA : Ax and SD : DF :: SA : Ax; .-. SE-SD : DE :: SA : Ax, or SO : OD :: SA : Ax, a relation analogous to SG : AC :: SA : AX. We have already shewn, for each conic, that the middle points of parallel chords lie in a straight line ; the following article contains a proof of the theorem which includes all the three cases. 185. Prop. II. To find the locus of the middle points of a system of parallel chords. 12—2 180 OF CONICS IN GENERAL. Let PT one of the chords he produced to meet the directrix in F, draw Ax parallel to FP, and divide FS so that SD : DF :: SE : EF :: SA : Ax; then, as in the preceding article, the perpendicular OV upon PP' from 0, the middle point of DE, hisects PP'. Draw the parallel focal chord aSa' ; then Oc parallel to the directrix hisects aa' in c. Also draw SG perpendicular to the chords, and meeting the directrix in 0. Then, if F meet aa in n, Vn : nO :: SF : SO, :: Sf : Sc. and, since ncO, SG/ are similar triangles, nO : nc :: SG : Sf; .-. Vn : nc :: SG : Sc, and the line Vc passes through G. The straight line Gc is therefore the locus of the middle points of all chords parallel to aSa'. The ends of the diameter GC may he found hy the construction of the preceding article. OF CONICS IN GENERAL. 186. When the conic is a parabola, SA = AX and Sa : af :: AX : Ax :: 8X : Sf. So Sa' : a'/ :: SX : Sf; .-. Sc : fflc :: SX : Sf, and ac : cf :: SX : Sf Hence Sc : cf :: SX^ : Sf :: GX .Xf: Gf.fX :: GX : Gf; 181 and therefore Gc is parallel to SX, that is, the middle points of parallel chords of a parabola lie in a straight line parallel to the axis. 187. Prop. III. To find the hcus of the middle points of all focal chords of a conic. Taking the case of a central conic, and referring to the figure of the preceding article, let Oc meet SO in N ; then cN : NS ■.fX : SX, and cN : NO : GX : CX; .: cN^ : SN-.NG : -.fX.GX-.SX.CX : SX' : SX . GX. Hence it follows that the locus of c is an ellipse of which 80 is the transverse axis, and such that the squares of its axes are as SX : GX, or (Cor. Art. 63) as BC" : AG\ Hence the locus of c is similar to the conic itself 182 EXAMPLES. EXAMPLES. 1. If an ordinate, PiVT', to the transverse axis meet the tangent at the end of the latus rectum in T, SP=TI^, and TP . TF = SNK 2. A focal chord PSQ of a conic section is produced to meet the directrix in E, and KM, KN are drawn through the feet of the ordinates PM, QN of P and Q. If KN produced meet PN produced in R, prove that PR=PM. 3. The tangents at P and Q, two points in a conic, intersect in T; if through P, Q, chords be drawn parallel to the tangents at § and P, and intersecting the conic in p and q respectively, and if tangents at ' p and q meet in T, shew that Tt is a diameter. 4. Two tangents TP, TP" are drawn to a conic intersecting the directrix in F, F'. If the chord PQ cut the directrix in R, prove that SF : SF' :: RF : RF" 5. The chord of a conic PP" meets the directrix in K, and the tangents at P and P' meet in 7"; if RKR', parallel to ST, meet the tangents in R and R', KR=KR'. 6. The tangents at P and P', intersecting in T, meet the latua rectum in D and /)'; prove that the lines through D and /)', re- spectively perpendicular to SP and SP', intersect in ST. 7. If P, Q be two points on a conic, and p, a two points on the directrix such that pq subtends at the focus half me angle subtended by PQ, either Pp and Qq or Pq and Qp meet on the curve. 8 A chord PP' of a conic meets the directrix in F, and from any point T in PP", TLL' is drawn parallel to SF and meeting SP, SP' in L and L' ; prove that the ratio of 8L or SL' to the distance of T from the directrix is equal to the ratio of SA : AX. 9. If an ellipse and an hyperbola have their axes coincident and proportional, points on them equidistant from one axis have the sum of the squares on their distances from the other axis constant. 10. If Q be any point in the normal PQ, QR the perpendicular on SP, and QM the perpendicular on PN, QR : PM :: SA : AX. EXAMPLES. 183 11. Given a focus of a conic section inscribed in a triangle, find the points where it touches the sides. 12. PSQ is any focal chord of a conic section ; the normals at F and Q intersect in K, and KN is drawn perpendicular to PQ; prove that PJV is equal to SQ, and hence deduce the locus of iV. 13. Through the extremity P, of the diameter PQ of an ellipse, the tangent TPT' is drawn meeting two conjugate diameters in T, T' From P, Q the lines PR, QR are drawn parallel to the same conjugate diameters. Prove that the rectangle under the semiaxes of the ellipse is a mean proportional between the triangles PQR and OTT'. 14. Shew that a conic may be drawn touching the sides of a. triangle, having one focus at the centre of the circumscribing circle, and the other at the orthocentre. 15. The perpendicular from the focus of a conic on any tangent, and the central radius to the point of contact, intersect on the directrix. 16. A£, AC aie tangents to a conic at S, and C, and DEGF is drawn from a point D in AC, parallel to AB and cutting the curve in E and F, and BC in Q ; shew that BG^=1)E.I)F. IV. A diameter of a parabola, vertex F, meets two tangents in D and E and their chord of contact is G, shew that F(^=ED.FE. 18. P and Q are two fixed points in a parabola, and from any other point R in the curve, RP, RQ are drawn cutting a fixed diameter, vertex E, in B and C; prove that the ratio of EB to EC is constant. 19. If the normal at P meet the conjugate axis in g, and ffk be perpendicular to SP, Pk is constant ; and if M, parallel to the trans- verse axis, meet the normal at P in I, kl is constant. 20. A system of conies is drawn having a common focus S and a common latus rectum LSL'. A fixed straight line through iS intersects the conies, and at the points of intersection normals are drawn. Prove that the envelope of each of these normals is a parabola whose fo«us hes on LSL', and which has the given line as tangent at the vertex. CHAPTER X. Ellipses as Roulettes and Gussettes. 188. If a circle rolls on the inside of the circumference of a circle of doiible its radius, any point in the area of the rolling circle traces out an ellipse. Let C be the centre of the rolling circle, E the point of contact. Then, if the circle meet in Q a fixed radius OA of the fixed circle, the angle ECQ is twice the angle EOA, and therefore the arcs EQ, EA are equal. Hence, when the circles touch at A, the point Q of the rolling circle coincides with A, and the subsequent path of Q is the diameter through A. Let P be a given point in the given radius CQ, and draw RPN perpendicular to OA, and PR' parallel to OA. Then, OQE being a right angle, EQ is parallel to RP and therefore CR = GP= CR', so that OR and OR' are constant. ELLIPSES AS ROULETTES AN0 GLISSETTES. Also PJff : RN" :: PQ : OR; 185 therefore, the locus of R being a circle, the locus of P is an ellipse, whose axes are as PQ : OR. But OR is clearly the length of one semi-axis, and PQ or OR' is therefore the length of the other, OR, OR' being equal to 00 + GP and OG - GP. 189. Properties of the ellipse are deducible from this construction. Thus, as the circle rolls, the point E is instantaneously at rest, and the motion of P is therefore at right angles to EP, i.e. producing EP to F, in the direction FO. Therefore, drawing PT parallel to OF, PT is the tangent, and PF the normal. The angles EPT, EQT being right angles, the points E, P, Q, T are concyclic ; but the circle through QPE clearly passes through R; therefore the angle ERT and consequently the angle ORT is a right angle, ON : OR :: OR : OT, OF. 0T=OR\ and or which is the theorem of Art. 74. Again, since EQ't and EPt are right angles, E, Q', t, P are concyclic; but the circle through EQ'P clearly passes 186 ELUPSES AS BOULETTES AND OLISSETTE& through R'; therefore the angle ER't and consequently the angle ORt is a right angle, and FN : OR' :: OK : Ot, FN . Ot = OR", or which is the theorem of Art. 75. Further, if FF meet OQ in G, the angles PQG, FFQ are equal, being on equal bases EQ, OQ' ; .-. FG : FQ :: FQ : FF, or FG .FF=FQ*=-OR'^, which is the first of theorems of Art. 77. And again, if FGF produced meet QO produced in g, the angles FQ'g, FFQ are equal, being on equal bases QO, EQ ; and the angle QFg is common to the two triangles Fgg.FFQ. Therefore these triangles are similar, and Fg : FQ :: FQ : FF, or Fg.FF=FQ\ But FQ = ER' = OR ; .-. Fg.FF=OR\ which is the second theorem of Ai't. 77. ELLIPSES AS BOUIjnTES ASD GLISSETTES. 187 190. If the carried point P is outside the circle the line PNR, perpendicular to OA, will meet OE produced in R, and GB will he equal to GF, so that OR will be constant and the locus of R will be a circle. Also, the triangles PQN, RON being similar, we shall have PJV : RN :: PQ : OR, so that the locus of P will be an ellipse, the semi-axes of which will be GP + 00 and GP - 00. 191. The fact that a point on the circumference of the rolling circle oscillates in a straight line is utilized in the construction of Wheatstone's Photometer. By help of machinery a metallic circle, about an inch in diameter, is made to roll rapidly round the inside of a circle of double this diameter, and carries a small bright bead which is fastened to its circumference. If this machine is held between two candles or other sources of light, 80 that the line of oscillation of the bead is equidistant from the candles, two bright lines will be seen in close contiguity, and it is easy to form an estimate of their comparative brightnesses. If bright beads are fastened to points in the area of the rolling circle not on the circumference, and the machine be held near sources of light, the appearance, when the circle is made to rotate rapidly, will be that of a number of bright concentric ellipses. 192. A given straight line has its ends moveahle on two straight lines at right angles to each other; the path of any given point in the moving line is an ellipse. 188 ELUPSES AS BOULETTES AND OLISSETTES. Let P be the point in the moving line AB, and G the middle point of AB. Let the ordinate NP, produced if necessary, meet 00 ia Q ; then CQ = CP and OQ = AP, so that the locus of Q is a circle. Also PN : QN :: PB : OQ :: PB : PA; therefore the locus of P is an ellipse, and its semi-axes are equal to AP and BP. 193. The theorem of Art. 188 is at once reducible to this case, for, taking the figure of Art. 189, QPQ' is a diameter of the rolling circle and is therefore of constant length, and the points Q and Q' move along fixed straight lines at right angles to each other ; the locus of P is therefore an ellipse of which Q'P and PQ are the semi-axes. 194. From this construction also properties of the tan- gent and normal are deducible. Complete the rectangle OAEB; then, since the direc- tions of motion of A and B are respectively perpendicular to EA and EB, the state of motion of the line AB may be represented by supposing that the triangle EAB is turning round the point E. Hence it follows that EP is the normal to the locus of P, and that PT perpendicular to EP is the tangent. Let OF, parallel to PT, meet EP i-a. F; then 0, F, B, E are concyclic ; .-. the angle PFB = EOB = PBG, and the triangles PGB, PFB are similar. Hence PG : PB :: PB : PF, or PG.PF=PB\ where PB is equal to the semi-conjugate axis. Similarly, by joining AF, it can be shewn that Pg.PF=PA\ g being the point of intersection of PC? and A 0. ELLIPSES AS BOULETTES AND aLISSETTES. 189 Again, since EPT, EBT are right angles, B, T, P, E are concyclic, and Q is clearly concyclic with B, P, E; so that TQE is a right angle. Hence OQN and OQT are similar triangles, and OK : OQ :: OQ : OT, or ON.OT=PA\ where PA is equal to the semi-transverse axis. 195. Observing that F, 0, A, E, B are concyclic, we have PF.PE=PA.PB; .: PE is equal to the semi-diameter conjugate to OP. This suggests a construction for the solution of the problem, Having given a pair of conjugate diameters of an ellipse, it is required to determine the position and magnitudes of the principal axes. Taking OP and OB as the given semi-conjugate diameters, draw PF perpendicular to OD, and, in FP produced, take PE equal to OD. Join OE, bisect it in C, and in CE take CQ equal to CP. Then OB, OA, drawn perpendicular and parallel to PQ, and meeting CP in B and A, will be the directions of the axes, and their lengths will be AP and PB. 190 ELLIPSES AS BOULETTES AND GLISSETTES. 196. If a given triangle AQB move in its own plane so that the extremities A, B, of its base AB move on two fixed straight lines at right angles to each other, the path of the point Qis an ellipse. If be the point of intersection of the fixed lines, and G the middle point of AB, the angles GOB, CBO are equal, so that, as AB slides, the line GB, and therefore also the line GQ, turns round as fast as GO, but in the contrary direction. Produce OG to P, making GP = GQ; then the locus of P is a circle the radius of which is equal to OG + GQ. There is clearly one position oi AB for which the points 0, G, and Q are in one straight line. Let OX be this straight line, and let OG, GQ, be any other corresponding positions of the lines; then, if GE is paraUel to OX, GE bisects the angle PGQ, and, draw- ing PQN and GL perpendicular to OX, hence QN=GL QN : PK PE, PN=^GL + PE, OC-GP : OG+GP :: OG-GQ : OG+GQ, and .'. the locus of Q is an ellipse of which the semi-axes are OG + GQ and OG - GQ. If the straight lines through A and B perpendicular to OA and OB meet in K, the point K is the mstantaneous centre of rotation. The normal to the path of Q is therefore QK and the tangent is the straight line through Q perpen- dicular to QK. 197. Elliptic Compaues. If two fine grooves, at right angles to each other, he made on the plane surface of a plate of wood or metal, and if two pegs, fastened to a straight rod, be made to move in these ELLIPSES AS ROULETTES AND GLIBSETTES. 191 grooves, then a pencil attached to any point of the rod will trace out an ellipse. By fixing the pencil at different points of the rod, we can obtain elli{)Bea of any eccentricity, but of dimensions limited by the lengths of the rod and the grooves. Bwrgtmi^s Ettiptograph. OE is a groove in a stand which can be fixed to the paper or drawing board, and OA, OB are rods jointed at A, so that the end B can slip along the groove, while AO turns round the fixed end 0. G is the middle point of AB, CD is a rod, the length of which is half that of AB, and the end D can slide along the groove. It follows that the angle ADB is always a right angle. A rod DP is taken of any convenient length, and, by means of a chain round the triangle ADC, is made to move so as to be always parallel to OA. If the end B be moved along the groove, the end P will trace out an ellipse of which is the centre, and the lengths of its semi-axes will be the length of DP and of the difference between the lengths of OA and DP. This can be seen by drawing a line OF perpendicular to OE, and producing DP to meet it in F. The motion will be that of a rod of length OA sliding between OE and OF. See Dyck, Katalog der mathematiscken Instrumente, Munohen, 1892. 192 MISCELLANEOUS PROBLEMS. MISCELLANEOUS PROBLEMS. I. 1. On a plane field the crack of the rifle and the thud of the ball striking the target are heard at the same instant ; find the locus of the hearer. 2. PQ, P'^ are two focal chords of a parabola, and PR, parallel to PQ', meets in Ji the diameter through Q ; prove that PQ.P'g = PS^. 3. CP and CJD are conjugate semi-diameters of an ellipse; PQ is a chord parallel to one of the axes ; shew that OQ is parallel to one of the straight lines which join the ends of the axes. 4. A line cuts two concentric, similar and similarly situated ellipses in P, Q, q, p. If the line move parallel to itself, PQ . Qp is constant. 5. The portion of a tangent to an hyperbola intersected between the asymptotes subtends a constant angle at the focus. 6. If a circle be described passing through any point i* of a given hyperbola and the extremities of the transverse axis, and the ordinate JS'P be produced to meet the circle in Q, the locus of Q is an hyperbola. 7. PQ is one of a series of chords inclined at a constant angle to the diameter AB oi a circle; find the locus of the intersection of AP, BQ. 8. If from a point T in the director circle of an ellipse tangents TP, TF be drawn, the line joining T with the intersection of the nonnals at P and P passes through the centre. 9. The points, in which the tangents at the extremities 'of the transverse axis of an ellipse are cut by the tangent at any point of the curve, are joined, one with each focus ; prove that the point of intersec- tion of the joining lines lies in the normal at the point. 10. Having given a focus, the eccentricity, a point of the ciu^e, and the tangent at the point, shew that in general two conies can b^ described. 11. A }>arabola is described with its focus at one focus of a given central conic, and touches the conio; prove that its directrix will touch a fixed circle. 12. The extremities of the latera recta of all conies which have a common transverse axis lie on two parabolas. 13. The tangent at a moveable point P of a conic intersects a fixed tangent in Q, and from S a straight line is drawn perpendicular to SQ and meeting in R the tangent at P; prove that the locus of R is a straight line. MISCELLANEOUS PBOBLEMS. 193 14. On all parallel chords of a circle a series of isosceles triangles are described, having the same vertical angle, and having their planes perpendicular to the plane of the circle. Find the locus of^ their vertices ; and find what the vertical angle must be in order that the locus may be a circle. 15. A series of similar ellipses whose major axes are in the same straight line pass through two given points. Prove that the major axes subtend right angles at four fixed points. 16. Prom the centre of two concentric circles a straight line is drawn to cut them in P and Q; through P and Q straight lines are drawn parallel to two given lines at right angles to each other. Shew that the locus of their point of intersection is an ellipse. 17. A circle always passes through a fixed point, and cuts a given straight line at a constant angle, prove that the locus of its centre is an hyperbola. 18. The area of the triangle formed by three tangents to a parabola is equal to one half that of the triangle formed by joining the points of contact. 19. If a parabola be described with any point on an hyperbola for focus and passing through the foci of the hyperbola, shew that its axis will be parallel to one of the asymptotes. 20. S and S being the foci, P a point in the ellipse, if ffP be bisected in L, and AL be drawn from the vertex cutting SP in Q, the locus of Q is an ellipse whose focus is S. 21. If the diagonals of a quadrilateral circumscribing an ellipse meet in the centre the quadrilateral is a parallelogram. 22. A series of ellipses pass through the same point, and have a common focus, and their major axes of the same length ; prove that the locus of their centres is a circle. What are the limits of the eccentricities of the ellipses, and what does the ellipse become at the higher limit 1 23. li S, IT he the foci of an hyperbola, ZL' any tangent inter- cepted between the asymptotes, SL . BX = CL . ZL'. 24. Tangents are drawn to an ellipse from a point on a similar and similarly situated concentric ellipse; shew that if P, Q be the points of contact. A, A' the ends of the axis of the first ellipse, the loci of the intersections of AP, A'Q, and of AQ, A'P are two ellipses similar to the given ellipses. 25. Draw a parabola which shall touch four given straight lines. Under what condition is it possible to describe a parabola touching five given straight lines? 26. A fixed hyperbola is touched by a concentric ellipse. If the curvatures at the point of contact are equal the area of the ellipse is constant. -D n o IS 194 MISCELLANEOUS PROBLEMS. 27. A circle passes through a fixed point, and cuts off equal chords AB, CD from two given parallel straight lines; prove that the en- velope of each of the chords AD, BC is a central conic having the fixed point for one focus. 28. A straight line is drawn through the focus parallel to one asymptote and meeting the other; prove that the part intercepted between the curve and the asymptote is one-fourth the transverse axis, and the part between the curve and the focus one-fourth the latus- rectmn. 29. PQ is any chord of a parabola, cutting the axis in Z ; R, R are the two points in the parabola at which this chord subtends a right angle : if RR be joined, meeting the axis in L', LL' will be equal to the latus-rectum. 30. If two equal parabolas have the same focus, tangents at points angularly equidistant from the vertices meet on the common tangent. 31. A parabola has its focus at ^S*, and PSQ is any focal chord, while PR, QQ' are two ckords drawn at right angles to PSQ at its extremities ; shew that the focal chord drawn parallel to PR is a mean proportional between PR and QQ. 32. With the orthocentre of a triangle as centre are described two ellipses, one circumscribing the triangle and the other touching its sides ; prove that these ellipses are sinular, and their homologous axes at right angles. 33. ABCD is a quadrilateral, the angles at A and C being equal ; a conic is described about ABCD so as to touch the circumscribing circle of ABC at the point B; shew that BD is a diameter of the conic. 34. The volume of a cone cut off by a plane bears a constant ratio to the cube, the edge of which is equal to the minor axis of the section. 35. A tangent to an ellipse at P meets the minor axis in t, and tQ is perpendicular to SP; prove that SQ is of constant length, and that if Pif be the perpendicular on the minor axis, QM will meet the major axis in a fixed point. 36. Describe an ellipse with a given focus touching three given straight lines, no two of which are parallel and on the same side of the focus. 37. Prove that the conic which touches the sides of a triangle, and has its centre at the centre of the nine-point circle, has one focus at the orthocentre, and the other at the centre of the circumscribing circle. 38. From Q, the middle point of a chord PR of an ellipse whose focus is S, QG is drawn perpendicular to PR to meet the major axis in G; piwe that 2.SG: SP+SR :: SA : AX. MISCELLAKEOXJS PROBLEMS. 195 39. A straight rod moves in any manner in a plane ; prove that, at any instant, the directions of motion of all its particles are tangents to a parabola. 40. If from a point T on the auxiliary circle, two tangents be drawn to an ellipse touching it in P and Q, and when produced meeting the circle again vap,q; shew that the angles PSp and QSq are together equal to the supplement of PTQ. 41. Tangents at the extremities of a pair of conjugate diameters of an ellipse meet in T; prove that ST, S'T meet the conjugate diameters in four concycUc points. 42. From the point of intersection of an asymptote and a directrix of an hyperbola a. tangent is drawn to the curve ; prove that the line joining the point of contact with the focus is parallel to the asymptote. 43. If a string longer than the circumference of an ellipse be always drawn tight by a pencil, the straight portions being tangents to the ellipse, the pencil will trace out a confocal ellipse. 44. 2) is any point in a rectangular hyperbola from which chords are drawn at right angles to each other to meet the curve. If P, Q be the middle points of these chords, prove that P, Q, D and the centre of the hyperbola are concyclic. 45. From a point T in the auxiliary circle tangents are drawn to an ellipse, touching it in P and §, and meeting the auxihary circle again in p and q ; shew that the angle pCq is equal to the sum of the angles PSQ and PSQ. 46. The angle between the focal distance and tangent at any point of an ellipse is half the angle subtended at the focus by the diameter through the point. 47. HisSk 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\a. P and §; 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. 48. Tangents to an ellipse, foci iS and H, at the ends of a focal chord PHP' meet the farther directrix in Q, Q'. The parabola, whose focus is S, and directrix PP, touches PQ, PQ', in Q, Q; it also touches the normals at P, P', and the minor axis, and has for the tangent at its vertex the diameter parallel to PP'. 49. <$ is a fixed point, and E a point moving on the arc of a given circle ; prove that the envelope of the straight line through E at right angles to SE is a conic. 50. A circle passing through a fixed point S cuts a fixed circle inP, and has its centre at 0; the hues which bisect the angle SOP all touch a conic of which (S is a focus. 13—2 196 MISCELLANEOUS PROBLEMS. 51. The tangent to an ellipse at P meets the directrix, corresponding to jS, in ^: through Z a straight line ZQR is drawn cutting the ellipse in §, iJ ; and the tangents at Q, R intersect (on SP) in T. Shew that a conic can be described with focus S, and directrix PZ, to pass through Q, R and T; and that TZ-mH. be the tangent at T. 52. TP, TQ are tangents to an ellipse at P and Q; one circle touches TP at P and meets TQ in Q and Q ; another touches TQ at Q and meets TP in P and P ; prove that PiQ and QP' are divided in the same ratio by the ellipse. 53. If a chord RPQV meet the directrices of an ellipse in R and F, and the circumference in P and Q, then RP and §F subtend, each at the focus nearer to it, angles of which the sum is equal to the angle between the tangents at P and Q. 54. Two tangents are drawn to the same branch of a rectangular hyperbola from an external point; prove that the angles which these tangents subtend at the centre are respectively equal to the angles which they make with the chord of contact. 55. If the normal at a point P of an hyperbola meet the minor axis in ff, Pg will be to /S'^ in a constant ratio. 56. An ordinate NP of an ellipse is produced to meet the auxiliary circle in Q, and normals to the ellipse and circle at P and Q meet in R ; RK, RL are drawn perpendicular to the axes; prove that KPL is a straight line, and also that KP=BC and LP=AC. 57. If the tangent at any point P cut the axes of a conic, produced if necessary, in T and T', and if C be the centre of the curve, prove that the area of the triangle TCT' varies inversely as the area of the triangle PCiV, where PN is the ordinate of P. 58. The circle of curvature of an ellipse at P passes through the focus S, SM is drawn parallel to the tangent at P to meet the diameter PCP in M; shew that it divides this diameter in the ratio of 3 : 1. 59. Prove the following construction for a pair of tangents from any external point 7" to an ellipse of which the centre is C: join CT, let TPCPT a similar and similarly situated ellipse be drawn, of which CT is a diameter, and P, P are its points of intersection with the given ellipse; TP, TP' will be tangents to the given ellipse. 60. Through a fixed point a pair of chords of a circle are drawn at right angles: prove that each side of the quadrilateral formed by joining their extremities envelopes a conic of which the fixed point and the centre of the circle are foci. 61. Any conic passing through the four points of intersection of two rectangular hyperbolas will be itself a rectangular hyperbola. 62. R is the middle point of a chord PQ of a rectangular hyperbola whose centre is C. Through R, RQ, RP' are drawn parallel to the tangents at P and Q respectively, meeting CQ, CP in §", P. Prove that C, P', R, Q' are concyclic. MISCELLANEOUS PROBLEMS. 197 63. The tangents at two points Q, ^ oi a. parabola meet the tangent at P in /J, R' respectively, and the diameter through their point of intersection T meets it in K; prove that PR=K£{!, and that, if QM, qM, TN be the ordinates of §, q, T respectively to the diameter through P, PR is a mean proportional between PM and PiT. 64. Common tangents are drawn to two psoabolas, which have a common directrix, and intersect in P, Q: prove that the chords joining the points of contact in each parabola are parallel to PQ, and the part of each tangent between its points of contact with the two curves is bisected by PQ produced. 65. An ellipse has its centre on a given hyperbola and touches the asymptotes. The area of the ellipse being tdways a maximum, prove that its chord of contact with the asymptotes always touches a similar hyperbola. 66. A circle and parabola have the same vertex A and a common axis. BA'C is the double ordinate of the parabola which touches the circle at A', the other extremity of the diameter which passes through A ; PP is any other ordinate of the parabola parallel to this, meeting the axis in N and the chord -45 produced in R : shew that the rectangle between RP and RP is proportional to the square on the tangent drawn from iV to the circle. 67. Tangents are drawn at two points, P, P on an ellipse. If any tangent be drawn meeting those at P, P in R, R', shew that the line bisecting the angle RSR' intersects RR' on a fixed tangent to the eUipse. Find the point of contact of this tangent. 68. Having given a pair of conjugate diameters of an ellipse, POP, BCD, let PF be the perpendicular from P on CD, in PF take PE equal to CD, bisect CE in 0, and on CE as diameter describe a circle ; prove that PO will meet the circle in two points § and R such that C§, CR are the directions of the semi-axes, and P§, PR their lengths. 69. A straight line is drawn through the ang^ular point .4 of a triangle ABC to meet the opposite side in a ; two points 0, & are taken on Aa, and CO, C€f meet AB in c and c', and BO, BO' meet CA in h, V ; shew that a conic passing through ahh'cd will be touched by BC. 70. If TP, TQ are two tangents to a parabola, and any other tangent meets them in Q and R, the middle point of QR describes a straight line. 71. Lines from the centre to the points of contact of two parallel tangents to a rectangular hyperbola and concentric circle make equal angles with either axis of the hyperbola. 72. A line moves between two lines at right angles so as to sub- tend a right angle and a half at a fixed point on the bisector of the right angle ; prove that it touches a rectangular hyperbola. 198 MISCELLANEOUS PROBLEMS. 73. Two cones, whose vertical angles are supplementary, are placed with their vertices coincident and their axes at right angles, and are cut by a plane perpendicular to a common generating line; prove that the directrices of the section of one cone pass through the foci of the section of the other. 74. The normal at a point P of an ellipse meets the curve again in P, and through 0, the centre of curvature at P, the chord ^0^ is drawn at right angles to PP ; prove that QO.Oq -.PO.OP -.-.i.PO-.PP. 75. From an external point T, tangents are drawn to an ellipse, the points of contact being on the same side of the major axis. If the focal distances of these points intersect in M and N, TM, TN are tangents to a confocal hyperbola, which passes through M and N. 76. Two tangents to an hyperbola from T meet the directrix in F and F ; prove that the circle, centre T, which touches SF, SF, meets the directrix in two points the radii to which from the point T are parallel to the asymptotes. 77. QR, touching the ellipse at P, is one side of the parallelogram formed by tangents at the ends of conjugate diameters ; if the normal at P meet the axes in O and g, prove that Q& and Rg are at right angles. 78. If PP be a double ordinate of an ellipse, and if the normal at P meet CP in 0, prove that the locus of is a similar ellipse, and that its axis is to the axis of the given ellipse in the ratio AC^-BC^ : AC^+BO. 79. A chord of a conic whose pole is T meets the directrices in R and R' ; if SR and S'R' meet in Q, prove that the minor axis bisects TQ. 80. On a parabola, whose focus is S, three points Q, P, Q are taken such that the angles PSQ, PSQ are equal ; the tangent at P meets the tangents at q, Q in T,T: shew that TQ : TQ ::SQ: SQ^. 81. If from any point i* of a parabola perpendiculars PJ^, PL are let fall on the axis and the tangent at the vertex, the Une ZiT always touches another parabola. 82. PQ is any diameter of a section of a cone whose vertex is V; prove that VP+ VQ is constant. 83. If SF, SK are the perpendiculars from a focus on the tangent and normal at any point of a conic, the straight line F£^ passes through the centre of the conic. 84. If the axes of two parabolas are in the same direction, their common chord bisects their common tangents. MXSCELLANEOUS PROBLEMS. 199 85. Find the position of the normal chord which cuts off from a parabola the least segment 86. From the point in which the tangent at any point i* of an hyperbola meets either asymptote perpendiculars PM, PN are let fall upon the axes. Prove that MN passes through P. 87. If two parabolas whose latera recta have a constant ratio, and whose foci are two given points S, S\ have a contact of the second order at P, the locus of P is a circle. 88. Find the class of plane curves such that, if from a fixed point in the plane, perpendiculars are let fall on the tangent and normal at any point of any one of the curves, the join of the feet of the perpen- diculars will pass through another fixed point. 89. If two ellipses have one common focus aS and equal major axes, and if one ellipse revolves in its own plane about S, the chord of intersection envelopes a conic confocal with the fixed ellipse. 90. The tangent at any point P of an ellipse meets the axis minor in T and the focal distances SP, HP meet it in R, r. Also ST, HT, Produced if necessary, meet the normal at P in Q, q, respectively, rove that Qr and qR are parallel to the axis major. 91. Two points describe the circumference of an ellipse, with velocities which are to one another in the ratio of the squares on the diameters parallel to their respective directions of motion. Prove that the locus of the point of intersection of their directions of motion will be an ellipse, confocal with the given one. 92. If AA' be the axis major of an elliptic section of a cone, vertex 0, and a AO, A'G' perpendicular to AV, ul'Fmeet the axis of the cone in G and Q', and QU, G'Vhe^ the perpendiculars let fall on AA', prove that Cand IT are the centres of curvature at A and A'. 93. By help of the geometry of the cone, or otherwise, prove that the sum of the tangents from any point of an ellipse to the circles of curvature at the vertices is constant. 94. If two tangents be drawn to a section of a cone, and from their intersection two straight lines be drawn to the points where the tangent plane to the cone through one of the tangents touches the focal spheres, prove that the angle contained by these lines is equal to the angle between the tangents. 95. If CP, CD are conjugate semi-diameters and if through C is drawn a line parallel to either focal distance of P, the perpendicular fi\)m D upon this line will be equal to half the minor axis. 96. The area of the parallelogram formed by the tangents at the ends of any pair of diameters of a central conic varies inversely as the area of the parallelogram formed by joining the points of contact 200 MISCELLANKOUS PROBLEMS. 97. Shew how to draw through a given point a plane which will have the given point for (1) focus, (2) centre, of the section it makes of a given right circular cone : noticing any limitations in the position of the point which may be necessary. 98. In the first figure of Art. 148, if a plane be drawn inter- secting the focal spheres in two circles and the cone in an ellipse, the sum or difference of the tangents from any point of the ellipse to the circles is constant. 99. If sections of a right cone be made, perpendicular to a given plane, such that the distance between a focus of a section and that vertex which lies on one of the generating lines in the given plane be constant, prove that the transverse axes, produced if necessary, of all sections will touch one of two fixed circles. 100. A sphere rolls in contact with two intersecting straight wires ; prove that its centre describes an ellipse. CHAPTER XI. Harmonic Properties, Poles and Polars. 198. Def. a straight line is harmonically divided in two points when the whole line is to one of the extreme parts as the other extreme part is to the m/iddle part. Thus AD\s harmonically divided in G and B, when AD : AC :: BD : BC. A d S i> This definition may also be presented in the following form. The straight line AB is harmonically divided in C and D, when it is divided internally in G, and externally in D, in the same ratio. Under these circumstances the four points A, G, B, D constitute an Harmonic Range, and if through any point four straight lines OA, OG, OB, OD be drawn, these four lines constitute an Harmonic Pencil. Prop. I. If a straight line be drawn parallel to one of the rays of an harmonic pencil, its segments made by the other three will be equal, and any straight line is divided harmoni- cally by the four rays. Let AGBD be the given harmonic range, and draw EGF through G parallel to OD, and meeting OA, OB in E and F. Then AD : AG :: OD : EG, and BD : BG :: OD : GF; 202 HARMONIC PKOPERTIES. but from the definition AD : AC :: BD : BG; .\EC=CF, and any other line parallel to ECF is obviously bisected by 00. Next, let achd be any straight line cutting the pencil, and draw ecf parallel to Od ; so that ec = cf. Then ad : ac :: Od : ec, and hd : he ■.: Od : cf; .'. ad : ac :: bd : be; that is, acbd is harmonically divided. If the line c^Sa be drawn cutting AO produced, then aS : ac :: OS : ec, and /3S : fie :: 08 : cf; .: oS : oc :: fiB : fie, or oc : aS :: fie : fiB, and similarly it may be shewn in all other cases that the line is harmonically divided. HASHONIC FBOFEBTIES. 203 199. Prop. II. The pencil formed by two straight lines and the bisectors of the angles between them is an harmonic pencil. For, if OA, OB be the lines, and 00, OD the bisectors, draw KPL parallel to 00 and meeting OA, OD, OB. Then the angles OKL, OLK are obviously equal, and the angles at P are right angles ; therefore KP = PL, and the pencil is harmonic. 200. Prop. III. If AGBD, Acbd be harmonic ranges, the straight lines Co, Bb, Dd will meet in a point, as also Cd, cD, Bb. 204 HARMONIC PROPERTIES. For, if Cc, Bd meet in F, join Fb; then the pencil F{Acbd) is harmonic, and will be cut harmonically by AD. Hence Fb produced will pass through B. Similarly, if Gd, cD meet in E, JS(Acbd) is harmonic, and therefore bE produced will pass through B. Harmonic Properties of a Quadrilateral. In the preceding figure, let GcdB be any quadrilateral; and let dc, DC meet in A, Cd, cD in E, and Cc, Dd in F. Then taking b and B so as to divide Acd and AGD har- monically, the ranges Acbd and AGBD are harmonic, and therefore Bb passes through both E and F. Similarly it can be shewn that AF is divided harmonically in L and M, by Dc and dG. For E(Acbd) is harmonic and therefore the transversal ALFM is harmonically divided. 201. Prop. IV. If AGBD be an harmonio range, and E the middle point of GD, EA.EB = EG\ A C B if D For AD : AC :: BD : BC, or AE + EC : AE-EG :: EG + EB : EC-EB; .: AE : EG :: EG : EB, or AE . EB = EC = ED". Hence also, conversely, if EC = ED^ = AE . EB, the range AGBD is harmonic, (7 and D being on opposite sides o{E. Hence, if a series of points A, a, B, 6,. ..on a straight line be such that EA .Ea = EB .Eb = EG. Ec... = EP', HAKMONIC PROPEETIES. 205 and a EQ = EP, then the several ranges {APaQ), (BPbQ), &c. are harmomc. 202. Def. a system of pairs of points on a straight line such that EA.Ea = EB .Eb=... =EP' = EQ', is called a system in Involution, the point E being called the centre and P, Q the foci of the system. Any two corresponding points A, a, are called conjugate points, and it appears from above that any two conjugate points form, with the foci of the system, an harmonic range. It will he noticed that a focus is a point ^t which conju- gate points coincide, and that the existence of a focus is only possible when the points A and a are both on the same side of the centre. 203. Prop, V. Having given two pairs of paints, A and a, B and b, it is required to find the centre and foci of the involution. If E be the centre, EA : EB :: Eb : Ea; ^ p Q :E a B * 6 a .: EA : AB :: Eb : ab, or EA : Eb :: AB : ab. This determines E, and the foci P and Q are given by the relations EP' = EQ* = EA.Ea. We shall however find the following relation usefiil. Since or but Again, Qb- EA : Eb : : EB : Ea; EA : Ab : : EB : aB, EA : EB: : Ab : aB; Eb ; ■.EA : : ab ■.AB; Eb : EB : : Ab. ba : AB.l Qb: : Pb : :QB ■.PB; -Pb ; : Pb : ■.QB- -PB : PB, Ba. 206 HABMONIC PROPERTIES. or 2.EP : Pb .: 2.EB : BP; :. Pb'' . PB' :: EP' : EB', Eb : EB Ab.ba : AB . Ba. This determines the ratio in which Bb is divided by P. 204. If QAPa be an harmonic range and E the middle point of PQ, and if a circle be described on PQ as diameter, the lines joining any point R on this circle with P and Q will bisect the angles between AR and aR. For EA . Ea = EF' = ER* ; .: EA : ER :: ER : Ea, and the triangles ARE, aRE are similar. Hence But AR aR : : EA ER : EA EP. Ea : EP : : EP EA; aP . EP : AP : EA. AR : aR : AP :aP, Hence and ARa is bisected by RP. Hence, if A and a, B and b be conjugate points of a system in involution of which P and Q are the foci, it follows that AB and ah subtend equal angles at any point of the circle on PQ as diameter. This fact also affords a means of obtaining the relations of Art, 203. HARMONIC PROPERTIES. 207 We must observe that if the points J., a are on one side of the centre and B, b on the other, the angles subtended by AB, ah are supplementary to each other. 205. Prop. VI. If four points form an harmonic range, their conjugates also form an harmonic range. Let A, B, G, I) he the four points, a, b, c, d their conjugates. d c b a A B C 1) e E p Then, as in the eighth line of Art. 203, EA : Ed :: AD : ad, . or ED : Ea :: AB : ad; .: AD .Ea = ED . ad. Similarly AO.Ea = EG.ac, BD.Eb=ED.bd. BC.Eb^EC.bc. But, ABGD being harmonic, AD : AG :: BD : BG; .: ED. ad : EG . ac :: ED . bd : EG . be. Hence ad : ac :: bd : be, or the range of the conjugates is harmonic. 206. Prop. VII. If a system, of conies pass through four given points, any straight line vnll he cut by the system in a series of points in involution. The four fixed points being C, D, E, F, let the line meet one of the conies in A and a, and the straight lines GF, ED, in B and b. Then the rectangles AB . Ba, GB . BF are in the ratio of the squares on parallel diameters, as also are Ah . ha and Db . bE. But the squares on the diameters parallel to GF, ED are in the constant ratio KF . KG : KE . KD ; and, the 208 HARMONIC PROPERTIES. line Bb being given in position, the rectangles CB . BF and Db . bE are given ; therefore the rectangles AB . Ba, Ab . ba are in a constant ratio. But (Art. 203) this ratio is the same as that of PB' to Pb", if P be a focus of the involution A, a, B, b. Hence P is determined, and all the conies cut the line Bb in points which form with B,b & system in involution. We may observe that the foci are the points of contact of the two conies which can be drawn through the four points touching the line, and that the centre is the intersection of the line with the conic which has one of its asymptotes parallel to the line. 207. Prop. VIII. If through any paint two tangents be drawn to a conic, any other straight line through the point will be divided harmonically by the curve and the chord of contact. Let AB, AG he the tangents, ADFE the straight line. Through D and E draw GBHK, LEMN parallel to BG. Then the diameter through A bisects BH, and BG, and therefore bisects QK ; hence GD = HK, and similarly LE = MK. Also LE : EN :: GD : DK; :. LE .EN : LE* :: GD . DK : GD", or LE.LM : GD . GH :: LE" : GD' :: LA' : GA\ HARMONIC PROPERTIES. But LE . LM : OD . OH :: LB" : BG'; hence AL : AG :: BL : BG, 200 and therefore AE : AD :: FE : FD, that is, ADFE is harmonically divided. 208. Prop. IX. If two tangents he drawn to a conic, any third tangent is hammnically divided by the two tangents, the curve, and the chord of contact. 14 210 HARMONIC FROPERTIES. Let DEFG be the third tangent, and through G, the point in which it meets AC, draw GHKL parallel to AB, cutting the curve and the chord of contact in H, K, L. Then GH . GL : GC :: AB' : AC :: GK' : GC; .'. GH .GL = GK\ Hence D(? : BE* :: GK' : EE' GH .GL : EBf FG' : FE' ■ that is, DEFG is an harmonic range. 209. Prop. X. Jf any straighi line meet two tangents to a conic in P and Q, the chord of contact in T and the conic in Rand V, PR.PV : QR. QV :: P2™ : Qr. Taking the preceding figure, draw the tangent DEFG parallel to PQ. Then and but 210. Prop. XI. If chords of a conic be drawn through a fixed point the pairs of tangents at their extremities will intersect in a fixed line. Let B be the fixed point and G the centre, and let CB meet the curve in P. Take A in CP such that CA : GP :: CP : CB; then B is the middle point of the chord of contact of the tangents AQ, AR. PR. PV: EF* : : PB' : BE' : DE'; QR QV: GF' : : QG' : Qr : GC : DG'; EF : DE : : GF : DG; PR .PV pr : : QR. QV : Qr UABMONIC PROPERTIES. 211 Draw any chord EBF, and let the tangents at E and F meet in G : also join CG and draw PN parallel to EF. Then if GG meet EF in K and the tangent at P in T, GK . GG=GN.GT; .: CG : GT GN GK GP GB GA CP; hence AG is parallel to PT, and the point G therefore lies on a fixed line. If the conic be a parabola, we must take AP equal to BP: then, remembering that KG and NT are bisected by the curve, the proof is the same as before. 211. If A. be the fixed point, let GA meet the curve in P, and take B in CP such that GB : CP :: CP : GA; then B is the middle point of the chord of contact of the tangents AQ, AR. 14—2 212 POLES AND POLAKS. Draw any chord AEF, and let the tangents at E and F meet in ; also join GG and draw PN parallel to EF. Then CK . GQ^CN . CT; :. CG : GT :: GN : GK :: GP : GA :: CB : GP; .: BG is parallel to PT and coincides with the chord of contact QR. Hence, conversely, if from points on a straight line pairs of tangents he drawn to a conic, the chords of contact will pass through a fixed point. Poles and Polars. 212. Def. The straight line which is the locus of the points of intersection of tangents at the extremities of chords through a fixed point is called the polar of the point. Also, if from points in a straight line pairs of tangents he drawn to a conic, the point in which all the chords of contact intersect is called the pole of the line. If the pole be without the curve the polar is the chord of contact of tangents from the pole. POLES AND POLARS. 213 If the pole be on the curve the polax is the tangent at the point. It follows at once from these definitions that the focus of a conic is the pole of the directrix, and that the foot of the directrix is the pole of the latus rectum. 213. Pkop. XII. A straight line drawn through any point is divided harmonically by the point, the curve, and the polar of the point. If the point be without the conic this is already proved in Art. 207. If it be within the conic, as B in the figure of Art. 210, then, drawing any chord FBEV meeting in V the polar of B, which is AG, the chord of contact of tangents from V passes through B, by Art. 211, and the line VEBF is there- fore harmonically divided. Hence the polar may be constructed by drawing two chords through the pole and dividing them harmonically; the line joining the points of division is the polar. Or, in the figure of Art. 210, GB.GA= GP\ so that the polar of B is obtained by taking the point A on the diameter through B, at the distance from C given by the above relation, and then drawing jIC parallel to the diameter which is conjugate to OP. Cor. Hence it follows that the centre of a conic is the pole of a line at an infinite distance. For, if GB is diminished indefinitely, GA is increased indefinitely. 214. PaOP. XIII. The polars of two points intersect in the pole of the line joining the two points. For, if A, B be the two points and the pole of AB, the line AO is divided harmonically by the curve, and therefore the polar of A passes through the point 0. 214 POLES AND POLARS. Similarly the polar of B passes through ; That is, the polars of A and B intersect iu the pole o{ AB. 215. Prop. XIV. If a quadrilateral be inscribed in a conic, its opposite sides and diagonals will intersect in three points smh that eaxih is the pole of the line joining the other two. Let ABCD be the quadrilateral, F and G the points of intersection of AD, BG, and of DC, AB. Let EG meet FA, FB, in L and M. Then (Art. 200) FDLA and FCMB are harmonic ranges ; Therefore L and M are both on the polar of F (Art. 213), and EG is the polar of F. Similarly, EF is the polar of G, and therefore E is the pole of i'C (Art. 214). 216. Def. If each of the sides of a triangle be the polar, with regard to a conic, of the opposite angular point, the triangle is said to be self-conjugate with regard to the conic. Thus the triangle EGF in the above figure is self-conju- gate. POLES AND POLARS. 215 To construct a self-conjugate triangle, take a straight line AB and find its pole C. Draw through G any straight line CD cutting AB in D, and find the pole E of CD, which lies on AB : then CDE is self-conjugate. 217. Prop. XV. If a quadrilateral circumscribe a conic, its three diagonals form a self-conjtigate triangle. Let the polar of F (that is, the chord of contact P'P), meet FG in M ; then, since B, is on the polar of F, it follows that F is on the polar of R. Now F (AEBG) is harmonic (Art. 200), and, if FE meet P'P in T, P'TPR is an harmonic range; hence, by the theorem of Art. 213, FT, i.e. FE, is the polar of R. Similarly, if the other chord of contsict QQ' meet FG in R', GE is the polar of R ; .: E is the pole of RR', that is, of LK. Again, DEBK is harmonic, and therefore the pencil C{QEPK) is harmonic. Hence, if QP meet AG in S and CK in V, QSPV is harmonic, and therefore S is on the polar of V. 216 POLES AND POLARS. But S is on the polar of (7; therefore CV, that is, CK, is the polar of S. Similarly, if P'Q' meet AC in S', AK is the polar of /S'. Hence it follows that K is the pole of SS', that is, of EL ; ELK is therefore a self-conjugate triangle. 218. Prop. XVI. If a system of conies have a comvwn self-conjugate triangle, any straight line passing through one of the angular paints of the triangle is cut in a series of points in involution. For, if ABC be the triangle, and a line APDQ meet BG in D, and the conic in P and Q, APDQ is an harmonic range, and all the pairs of points P, Q form with A and D an har- monic range. Hence the pairs of points form a system in involution, of which A and D are the foci. 219. Prop. XVII. The pencil formed by the polars of the four points of an harmonic range is an harmonic pencil. Let ABCD be the range, the pole of AD. Let the polars Oa, Oh, Oc, Od meet AD in a, h, c, d, and let AD meet the conic in P and Q. d b c a B I> Then APaQ, CPcQ, &c. are harmonic ranges ; and there- fore (Arts. 201, 202) a, c, b, d are the conjugates of A, G, B, D. Hence (Art. 205) the range achd is harmonic, and there- fore the pencil (acbd) is harmonic. EXAMPLES. 217 EXAMPLES. 1. If PSiJ.' is a right angle, S being the centre of the auxiliary circle. And, similarly, if SQ, SB, perpendiculars to SB', SC, meet G'A' and A'R in Q and R, these points correspond to BE and GF. But AD, BE, GF are concurrent ; .". P, Q, R are collineai. Hence the reciprocal theorem, If from any point S lines be drawn perpendicular respect- ively to SA', SF, SG', and tneeting EG', G'A', A'B' in P, Q, and R, these points are collinear. As a second illustration take the theorem. If A, B be two fixed points, and AG, BG at right angles to each other, the locus of G is a circle. Taking 0, the middle point of AB, as the centre of the auxiliary circle, the reciprocals of A and B are two parallel straight lines, PE, QF, perpendicular to AB ; the reciprocals of A G, BG are points P, Q on these lines such that POQ is a right angle, and PQ is the reciprocal of G. 222 KEOIPBOCAL POLABS. Hence, the locus of G being a circle, it follows that PQ always touches a circle. The reciprocal theorem therefore is, If a straight line PQ, bounded by two parallel straight lines, subtend a right angle at a point 0, halfway between the lines, the line PQ always Umches a circle, having for its centre. 223. Prop. II. The reciprocal polar of a circle with regard to another circle, called the auadliary circle, is a conic, a focus of which is the centre of the auxiliary circle, and the corresponding directrix the polar of the centre of the recipro- cated circle. Let S be the centre of the auxiliary circle, and KX the polar of G, the centre of the reciprocated circle. Then, if P be the pole of a tangent QF to the circle C, SP meeting this tangent in F, SP.8Y=SX.SC. Therefore, drawing SL parallel to QY, SP : SG :: SX : QL. RECIPROCAL POLARS. 223 But, by similar triangles, SP : SO :: SAT : GL; .•.SP:SG::NX:CQ, or SP : PK :: 80 : GQ. Hence the locus of P is a conic, focus S, directrix KX, and having for its eccentricity the ratio of SO to CQ. The reciprocal polar of a circle is therefore an ellipse, parabola, or hyperbola, as the point S is within, \ipon, or without the circumference of the circle. 224. Prop. III. To find the latus rectum and axes of the reciprocal conic. The ends of the latus rectum are the poles of the tan- gents parallel to SO. Hence, if 8R be the semi-latus rectum, SR.OQ=SE\ SE being the radius of the auxiliary circle. The ends of the transverse axis A, A' are the poles of the tangents at F and Q ; ■.SA.SG = SE\ and 8A'.SF = SE\ 224 RECIPBOCAL POLABS. Let SU, 8U' be the tangents from 8, then 8G . 8F=SU'', .■.SA'-.SG-.-.SE'-.SU") ,. and >Sf^ : SF :: 8E' : SU'] ^"-'' Hence A A' : FG :: SE^ : SIT, or, if be the centre of the reciprocal, AO-.CQ:: SE' : SCP. Again, if BOB' be the conjugate axis, BO'^SR.AO; therefore, since SE' = 8R . CQ, BO' : SE" :: AO : CQ :: ,Sf^' : SIP and B0.SU=8E\ The centre 0, it may be remarked, is the pole of UV. For, from the relations (a), SE' : SIP :: SA + SA' : 8F + SG 80: SG 80 .SM:80. SM: .•.S0.SM=SE'. 225. In the figures drawn in the two preceding articles, the reciprocal conic is an hyperbola; the asymptotes are therefore the lines through perpendicular to 8U a,nd SU', the poles of these lines being at an infinite distance. The semi-conjugate axis is equal to the perpendicular from the focus on the asymptote (Art. 103), i.e. if OD be the asymptote, SD is equal to the semi-conjugate axis. Further, since OD is perpendicular to SU, and is the pole of UU', it follows that I) is the pole of CU, and that SD.8U=SE', as we have already shewn. RECIPROCAL POLARS. 225 Again, D, being the intersection of tLe polars of C and IT, is the intersection of SU and the directrix. 226. If the point S be within the circle, so that the reciprocal is an ellipse, the axes are given by similar rela- tions. Through 8 draw £fF perpendicular to FQ, and let UMV be the polar of S with regard to the circle. Then SM.8C=SC.CM-S(P = Cr-80* = 8Tr'; also, SE being the radius of the auxiliary circle, SA.8F=8E' = SA'.8G, and SF.SG = 8r'; .-. 8A : 8G :: SE* : 8V\ 8A' : 8F :: SE* : 8ry Hence 80 : 80 :: 8E' : 8V, and 80.8M:SC. 8M :: 8E* : 8T; .■.S0.8M = 8E', so that is the pole of UU'. Again 8A + SA' : 8F+ 8G :: 8E* : SV\ .: AO : CQ :: 8E' : 8V\ If B8Bf is the latus-rectum, 8R.CQ=8E\ B. c. s. 15 226 RECIPKOCAI- FOLABS. and if BOlff is the minor axis SR.AO = BO*; .: BO* :SE':: SE' : ST, and BO.Sr = SE\ 227. The important Theorem we have just considered enables us to deduce from any property of a circle a cor- responding property of a conic, and we are thus furnished with a method, which may serve to give easy proofs of known properties, or to reveal new properties of conies. In the process of reciprocation we observe that points become lines and lines points; that a tangent to a curve reciprocates into a point on the reciprocal, that a curve inscribed in a triangle becomes a curve circumscribing a triangle, and that when the auxiliary curve is a circle, the reciprocal of a circle is a conic, the latus rectum of which varies inversely as the radius of the circle. Also, conversely, the reciprocal of a conic with regard to a circle having its centre at a focus of the conic is a circle the centre of which is the reciprocal of the directrix of the conic. For an ellipse the centre of reciprocation is within the circle, for a parabola it is upon the circle, and for an hyperbola it is outside the circle. 228. We give some transformations of theorems as illustrations of the preceding articles. Theobeu. Reciprocal. The line joining the points of The tangents at the ends of a contact of parallel tangents of a focal chord intersect in the direc- circle passes through the centre. trix. The angles in the same seg- If a moveable tangent of a ment of a circle are equal. conic meet two fixed tangents, the intercepted portion subtends a constant angle at the focus. Two of the common tangents If two conies have the same of two equal circles are paraUel. focus, and equal latera recta, the straight line joining two of their common points passes through the focua. RECIPKOCAL POLAItS. 227 Theorem. The tangent at any point of a circle is perpendicular to the diameter through the point. A chord of a circle is equally inclined to the tangents at ite ends. If a chord of a circle subtend a constant angle at a fixed point on the curve, the chord always touches a circle. If a chord of a circle pass through a fixed point, the rect- angle contained by the segments is constant. If two chords be drawn from a fixed point on a circle at right angles to each other, the line joining their ends passes through the centre. If a circle be inscribed in a triangle, the lines joining the ver- tices with the points of contact meet in a point. The sum of the reciprocals of the radii of the escribed circles of a triangle is equal to the reciprocal of the radius of the inscribed circle. The common chord of two in- tersecting circles is x>erpendicular to the line joining their centres. If circles pass through two fixed points, the locus of their centres is a straight line. Two tangents to a conic at right angles to each other inter- sect on a fixed circle. Hecipbocal. The portion of the tangent to a conic between the point of con- tact and the directrix subtends a right angle at the focus. The tangents drawn from any point to a conic subtend equal angles at a focus. If two tangents of a conic move so that the intercepted portion of a fixed tangent subtends a con- stant angle at the focus, the locus of the intersection of the moving tangents is a conic having the same focus and directrix. The rectangle contained by the perpendiculars from the focus on two parallel tangents is constant. If two tangents of a conic move so that the intercepted por- tion of a fixed tangent subtends a right angle at the focus, the two moveable tangents meet in the di- rectrix. If a triangle be inscribed in a conic the tangents at the vertices meet the opposite sides in three points lying in a straight line. With a given point as focus, four conies can be drawn circum- scribing a triangle, and the latus rectum of one is equal to the sum of the latera recta of the other three. If two parabolas have a com- mon focus, the hne joining it to the intersection of the directrices is perpendicular to the common tangent. If conies have a fixed focus and a pair of fixed tangents in common, the corresponding directrices all pass through a fixed point Chords of a circle which sub- tend a right angle at a fixed point all touch a conic of which that point is a focus. 15—2 228 KECIPBOCAL FOLAJEtS. 229. Prop. IV. A system of coaxal circles can he reciprocated into a system ofconfocal conies. Let X be the point at which the radical axis crosses thie line of centres, and let E and S be the limiting points of the system. Then XE is equal to the length of the tangent XD to any one of the circles, and, therefore, if A is the centre of this circle, AD is the tangent at D to the circle whose centre is X and radius XE. Hence it follows that AE .AS = AD'', shewing that UIF, the polar of S with regard to the circle A, passes through E. Reciprocating with regard to S, the centre of the re- ciprocal curve is the pole of UW, and is consequently fixed; and the conies are therefore confocal. Hence, if we reciprocate with regard to either limiting point, we obtain confocal conies. In the particular case in which the circles all touch the radical axis, we obtain confocal and coaxial parabolas. 230. Prop. V. The reciprocal polar of a conic with regard to a circle, or with regard to any conic, is a conic. Taking any two tangents of the conic, their reciprocal polats are points on the reciprocal curve, and the reciprocal polar of their point of intersection is the chord joining the points. Since only two tangents can be drawu from a point to a conic, it follows that the reciprocal curve is always intersected by a straight line in two points only. It follows therefore that the reciprocal curve is a conic. In reciprocating a conic with regard to a circle, the reciprocal polar is an ellipse, parabola, or hyperbola, according as the centre 8 of the circle is inside, upon, or outside the conic. In the second case the axis of the parabola is parallel to the normal at the point S, and in the third case the asymptotes are perpendicular to the tangents which can be diuwn from the point jS to the conic. RECIPROCAL POLARS. 229 When the auxiliary curve is a conic, centre S, the first of the preceding statements holds good. When the point S is on the conic, the axis of the parabola is parallel to the diameter of the auxiliary conic, vrhich is conjugate to the tangent at S. When the point S is outside the conic, the asymptotes of the hjrperbola are parallel to those diameters of the auxiliary conic which are conjugate to the straight lines through 8 touching the conic to be reciprocated. The follomng cases will serve to illustrate the theorem of this article. 231. The reciprocal polar of a parabola with regard to a point on the directrix is a rectangular hyperbola. For the two tangents from the point are at right angles to each other, and therefore the asymptotes are at right angles to each other. 232. The reciprocal polar of an ellipse or hyperbola, with regard to its centre, is a similar curve turned through a right angle about the centre. If OY is the perpendicular on the tangent at P, and Q the reciprocal of the tangent, CQ . CT is constant. But CT . CD is constant ; .•. CQ varies as CD, and the reciprocal curve is the same as the original curve, or similar to it. 233. The chords of a conic which subtend a right angle at a fixed point P of a conic all pass through a fixed point in the normal at P. Reciprocating with regard to P, the reciprocal curve is a parabola, the axis of which is parallel to the normal to the conic, and the reciprocal of the chord is the point of inter- section of tangents at right angles to each other. The locus of this point is the directrix of the parabola, and, being at right angles to the normal, it follows, on 230 RECIPROCAL POLARS. reciprocating backwards, that the chord passes through a fixed point E in the normal. To find the position of the point E, let C be the centre of the conic, CA, CB its semi-axes, and PNP' the double ordinate, and let the normal meet the axes in G and g. Since GA and CB bisect the angle PGF and its supplement, G{BPAP') is an harmonic pencil ; .•. PGEg is an harmonic range, so that PE is the harmonic mean between PG and Pg. In the case of an hyperbola EGPg is an harmonic range. In the case of a parabola, E is the point of intersection of the normal with the diameter through P'. 234. The chords of a conic which subtend a right angle at a fixed point not on the conic all touch a conic of which that point is a focus. Reciprocating with regard to 0, the reciprocal of the envelope of the chords is the director circle of a conic, and therefore, reciprocating backwards, it follows that the en- velope of the chords is a conic of which is a focus. This of course includes the preceding theorem as a particular case, the fact being that when is on the conic the envelope of the chords is a conic, with a vertex and focus at E, flattened into a straight line. 235, If the sides of a triangle are tangents to a parabola, the orthocentre of the triangle is on the directrix of the parabola. This theorem is at once obtained by reciprocating, with regard to the orthocentre of the triangle, the theorem, proved in Art. 143, that, if a rectangular hyperbola passes through the angular points of a triangle, it also passes through the orthocentre of the triangle. EXAMPLES. 231 EXAMPLES. 1. If any triangle be reciprocated with regard to its orthocentre, the reciprocal triangle will be similar and similarly situated to the original one and will have the same orthocentre. 2. If two conies have the same focus and directrix, and a focal chord be drawn, the four tangents at the points where it meets the conies intersect in the same point of the directrix. 3. An eUipse and a parabola have a common focus ; prove that the ellipse either intersects the parabola in two points, and has two common tangents with it, or else does not cut it. 4. Prove that the reciprocal polar of the circumscribed circle of a triangle with regard to the inscribed circle is an eUipae, the major axis of which is equal in length to the radius of the inscribed circle. 5. Reciprocate with respect to any point S the theorem that, if two points on a circle be given, the pole of PQ with respect to that circle lies on the line bisecting PQ at right angles. 6. If two parabolas whose axes are at right angles have a common focus, prove that the part of the common tangent intercepted between the points of contact subtends a right angle at the focus. 7. The tangent at a moving point P of a conic intersects a fixed tangent in Q, and from S a straight line is drawn perpendicular to SQ and meeting in R the tangent at P ; prove that the locus ot R ia a, straight line. 8. Four parabolas having a common focus can be described touching respectively the sides of the triangles formed by four given points. 9. A triangle ABC circumscribes a parabola, focus S; through ABG lines are drawn respectively perpendicular to SA, SB, SC ; shew that these lines are concurrent. 10. Prove that the distances, from the centre of a circle, of any two poles are to one another as their distances from the alternate polars. 11. Reciprocate the theorems, (1) The opposite angles of any quadrilateral inscribed in a circle are equal to two right angles. (2) If a line be drawn from the focus of an eUipse making a constant angle with the tangent, the locus of its intersection with the tangent is a circle. 232 EXAMPLES. 12. The locos of the intersection of two tangents to a parabola which include a constant angle is an hyperbola, having the same focus and directrix. 13. Two ellipses having a common focus cannot intersect in more than two real points, but two hyperbolas, or an ellipse and hyperbola, may do bo. 14. ABC is any triangle and P any point: four conic sections are described with a given focus touching the sides of the triangles ABC, PBC, PC A, PAB respectively ; shew that they all have a common tangent. 15. TP, TQ are tangents to a parabola cutting the directrix respectively in X and T ; ESF is a s^aight line drawn through the focus 5 perpendicular to ST, cutting TP, TQ respectively in JE, F; prove that the lines ET, XF are tangents to the parabola. 16. With the orthocentre of a triangle as focus, two conies ore described touching a side of the triangle and having the other two sides as directrices respectively ; shew that their minor axes are equal 17. Two parabolas have a common focus S ; parallel tangents are drawn to them at P and Q intersecting the common tangent in P" and ^; prove that the angle PSQ is equsd to the angle between the axes, and the angle P'SQ[ is supplementiuy. 18. ABC is a given triangle, S a given point; on BC, CA, AB respectively, points A', ff, C are taken, such that each of the angles ASA', BSff, CSC, is a right angle. Prove that A', B, C lie in the same straight line, and that the latera recta of the four conies, which have S for a common focus, and respectively touch the three sides of the triangles ABC, ABC, A' EC, A! EC are equal to one another. 19. A parabola and hyperbola have the same focus and directrix, and SPQ is a line drawn through the focus S to meet the parabola in P, and the nearer branch of the hyperbola in Q ; prove that i'Q varies as the rectangle contained by SP and SQ. 20. If two equal parabolas have the same focus, the tangents at points angularly equidistant from the vertices meet on the common tangent. 21. If an ellipse and a parabola have the same focus and directrix, and if tangents are drawn te the ellipse at the ends of its major axis, the diagonals of the quadrilateral formed by the four points where these tangents cut the parabola intersect in the focus. 22. Find the reciprocals of the theorems of Arts. 215 and 217. 23. If a conic be reciprocated with regard to a point, shew that there are only two positions of the point, such that the conic may be similar and similarly situated to the reciprocal TffYAMPT.TnS , 233 24. Conies are described having a common focus and equal latera recta. Also the corresponding directrices envelope a fixed confocal conic Prove that these conies all touch two fixed conies, and that the reciprocals of the latera recta of these fixed conies are equal to the sum and difference of the latera recta. of the variable conies and of the fixed confocaL 25. Given a point, a tangent, and a focus of a conic, prove that the envelope of the directrix is a conic passing through the given focus. 26. Two conies have a common focus : their corresponding direc- trices will intersect on their common chord, at a point whose focal distance is at right angles to that of the intersection of their common tangents. If the conies are parabolas, the inclination of their axes will be the angle subtended by the common tangent at the common focus. 27. If the intercept on a given straight line between two variable tangents to a conic subtends a right angle at the focus of the conic, the tangents intersect on a conic. 28. The tangent at P to an hyperbola meets the directrix in Q ; another point R is taken on the directrix such that QR subtends at the focus an angle equal to that between the transverse axis and an asymptote ; prove that RP envelopes a parabola. 29. S is the focus of a conic; P, Q two points on it such that the angle PSQ is constant ; through S, SR, ST are drawn meeting the tangents a.t P,Qm R, T respectively, and so that the angles PSR, QST are constant ; shew that RT always touches a conic having the same focus and directrix as the original conic. 30. OA, OB are common tangents to two conies having a common focus ;Si, CA, CB are tangents at one of their points of intersection, BD, AE tangents intersecting CA, CB, in 2), E. Prove that SHE is a straight line. 31. An hyperbola, of which S is one focus, touches the sides of a triangle ABC; the lines SA, SB, SC are drawn, and also lines SD, SE, SF respectively perpendicular to the former three lines, and meeting any tangent to the curve in D, E, F ; shew that the lines AD, BE, CF ajB concurrent. 32. If a conic inscribed in a triangle has one focus at the centre of the circumscribed circle of- the triangle, its tran.sverse axis is equal to the radius of that circle. 33. If any two diameters of an ellipse at right angles to each other meet the tangent at a fixed point P in Q and R, the other two tangents through Q and R intersect on a fixed straight Une which passes through a point T on the tangent at P, such that PCT is a right angle. CHAPTER XIII. The Construction of a Conic from Given Conditions. 236. It will be found that, in general, five conditions are sufficient to determine a conic, but it sometimes happens that two or more conies can be constructed which will satisfy the given conditions. We may have, as given conditions, points and tangents of the curve, the directions of axes or conjugate diameters, the position of the centre, or any characteristic or especial property of the curve. Pkop. I. To construct a parabola, passing through three given points, and having the direction of its aans given. In this case the fact that the conic is a parabola is one of the conditions. Let P, Q, R be the given points, and let RE parallel to the given direction meet PQ in E. CONSTRUCTION OF A CONIC FROM GIVEN CONDITIONS. 235 If E be the middle point of PQ, R is the vertex of the diameter RE; but, if not, bisecting PQ in V, draw the diameter through V and take A such that AV : RE :: QV' : QE.EP. Then A is the vertex of the diameter A V. If the point E do not fall between P and Q, A must be taken on the side of PQ which is opposite to R. The focus may then be found by taking A U such that qV* = 4iAV.AU, and by then drawing US parallel to QV and taking AS equal to A U. 237. Prop. II. To describe a parabola through /our given points. First, let ABGD be four points in a given parabola, and let the diameter CF meet AD in F. Draw the tangents PT, QT parallel to AD, BC, and the diameter QV meeting PT in V. 236 CONSTRUCTION OF A CONIC Then ED.EA : EG.EB TV* : TQ' EF' : EC. Hence the construction ; in EA take EF such that EF* : EC* :: ED.EA : EG.EB, then CF is the direction of the axis, and the problem is reduced to the preceding. If the point F be taken in AE produced, another para- bola can be drawn, so that, in general, two parabolas can be drawn through four points. 238. This problem may be treated differently by help of the theorem of Art. 52, viz. ; If from a point 0, outside a parabola, a tangent OM, and a chord OAB he dravm, amd if the diameter ME meet the chord in E, OE* = OA.OB. Let A, B, G, D be the given points, and let E, E', F, F', be so taken that OE' = OE'*==OA.OB. and 0F'=0F'* = OG.OR FBOM GIVEN CONDITIONS. 237 Then EF and E'F' are diameters, and KL, the polar of 0, will meet EF and E'F' in M, N, the points of contact of tangents from 0. The second parabola is obtained by taking for diameters EF' and E'F. 239. Pbop. III. Any conic passing through four points has a pair of conjugate diameters parallel to the axes of the two parabolas which can be drawn through the four points. Let TP, TQ be the tangents parallel to OAB and OCL, and such that the angle PTQ is equal to A OC. Then, if OE' = OA . OB, and OF^^OG . OD, OE' : OF' :: OA.OB : OC.OD .-. EF is parallel to PQ. Hence, if R and V be the middle points of EF and PQ, OR is parallel to TV; But, taking OF' equal to OF, OR is parallel to EF', .: TV and PQ are parallel to EF' and EF; i.e. the conjugate diameters parallel to TV and PQ are parallel to the axes of the two parabolas. 238 CONSTBUCTION OF A CONIC 240. Peop. IV. Having given a pair of conjugate dia- meters, POP", DCiy, it is required to construct the ellipse. In CP take E such that PE.PO= CD', draw PF per- pendicular to CJ), and take FC equal to FG. About CEG' describe a circle, cutting PF in G and G' ; then PG.PG' = PE.PG=GD', and GGG' is a right angle ; therefore CG and CG' are the directions of the axes and their lengths are given by the relations, PG .PF = BC, PG'.PF = AG^. We may observe that, being the centre of the circle, AC* + BG' = PF.PG + PF. PG' = 2.PF. PC = 2.PG.PN, if JV be the middle point of CE, '^PC'+PG.PE = CP' + ciy. FKOM GIVEN CONDITIONS. 239 If PE' be taken equal to PE in GP produced, and the same construction be made, we shall obtain the axes of an hyperbola having GP, GD for a pair of conjugate semi- diameters. 241. This problem may be treated also as follows. In PF, the perpendicular on GD, take PK =PE'=GD; then PK*=PO.PG', and therefore K'GKG' is an harmonic range ; and GGG' being a right angle, it follows (Art. 199), that GG and GG' are the bisectors of the angles between GK and GE'. Hence, knowing GP and GD, G and G' are determined. 242. Prop. V. Having given the/ocua and three points of a conic, to find the directrix. Let A, B, G, 8 be the three points and the focus. Produce BA to D so that BD : AD :: SB : SA, and GB to E, so that BE : CE :: SB : SG; then DE is the directrix. The lines BA, BG may be also divided internally in the same ratio, so that four solutions are generally possible. Conversely, if three points A, B, G and the directrix are given, let BA, BG meet the directrix in D and E; then 8 Hes on a circle, the locus of a point, the distances of which from A and B are in the ratio of AD to DB. 8 lies also on a circle, similarly constructed with regard to BGE; the intersection of these circles gives two points, either of which may be the focus. 240 CONSTBUCnON OF A CONIC 243. Prop. VI. Having given the centre, the directions of a pair of conjugaie diameters, and two points of an ellipse, to describe the ellipse. If be the centre, GA, CB the given directions, and P. Q the points, draw QM^, PLP' parallel to CB and CA, and make Q'M= QM and P'i = PL. Then the ellipse will evidently pass through P" and Q', and if CA, CB be the conjugate radii, their ratio is given by the relation CA' : CB' :: EP.EF : EQ.EQ, E being the point of intersection of P'P and Q'Q. Set up a straight line ND perpendicular to GA and such that ND" : NP' :: EP.EF : EQ. EQ, and describe a circle, radius CD and centre G, cutting CA in il, and take CB : CA :: JVP : ND. Then AN.NA''=ND\ and PiT : ^iV. NA' :: CF' : GA\ Hence CA, CB are determined, and the elhpse passes through P and Q. FBOU GIVEN CONDITIONS. 241 244. Pbop. VII. To describe a conic passing thrc/ugh a given point and tofuching two given straight lines in given points. Let OA, OB be the given tangents, J. and £ the points of contact, JV" the middle point of AB. 1st. Let the given point D be in OJV; then, if ND = OB, the curve is a parabola. But if i\rD < OD, the curve is an ellipse, and, taking C such that DC . CN = GJy, the point G is the centre. If ND > OD, the curve is an hyperbola, and its centre is found in the same manner. 2nd. If the given point be E, not in ON, draw GEF parallel to AB, and make FL equal to EL. Take K such that GK'=GE.GF; then AK produced will meet ON in D, and the problem is reduced to the first case. To justify this construction, observe that, if DM be the tangent at D, GE.GF : GA* :: DM^ : MA' :: GK* : GA\ so that GE.GF =GIP. B. c. s. 16 242 (CONSTRUCTION OF A CONIC 245. Prop. VIII. To draw a conic through five given points. Let A,B,C,D, E be the five points, and F the inter- section of DE, AB. Draw CO, CH, parallel respectively to AB and ED, and meeting ED, AB in G and H. If F and O fall between D and E, and F and H between A and B, take GP in C(? produced and HQ in C^ produced, such that CG.QP : DG. GE :: AF.FB : DF. FE, and C7S..ffQ : AH.HB :: DF.FE : AF.FB; Then (Arts. 92 and 134) P and Q are points in the conic. Also PC, AB being parallel chords, the line joining their middle points is a diameter, and another diameter is ob- tained from GQ and DE. If these diameters are parallel, the conic is a parabola, and we fall upon the case of Prop. II. ; but if they intersect in a point 0, this point is the centre of the conic, and, having the centre, the direction of a diameter, and two ordinates of that diameter, we fall upon the case of Prop. VI. FBOH GITEN COXDITIONS. 243 The figure is drawn for the case in which the pentagon AEBGD is not re-entering, in which case the conic may be an ellipse, a parabola, or an hyperbola. If any one point fall within the quadrilateral formed by the other four, the curve is an hyperbola. In all cases the points P, Q must be taken in accordance with the following rule. The points G, P, or 0, Q must be on the same or different sides of the points G, or H, according as the points D, E, or B, A are on the same or different sides of the points G or H. Thus, if the point E be between B and F, and if G be between D and E, and H between A and B, the points P and G will be on the same side of G, and G, Q on the same side of H, but if H do not fall between A and B, G and Q will be on opposite sides of H. Bemembering that if a straight line meet only one branch of an hyperbola, any parallel line will meet only one branch, and that if it meet both branches, any parallel will meet both branches, the rule may be established by an examination of the different cases. 246. The above construction depends only on the ele- mentary properties of Conies, which are given in Chapters I, II, III, and IV. For some further constructions we shall adopt another method depending on harmonic properties. Prop. IX. Hairing given two pairs of lines OA, OA', a/nd OB, OB, to find a pair of lines OG, OG', which shall make with each of the given pairs an harmonic pencil. This is at once effected by help of Art. 203. For, if any transversal cut the lines in the points c, a, h, c', h', a', the points c, c' are the foci of the involution, in which a, a' are conjugate, and also h, h', the centre of the involution being the middle point of cc'. 16—2 244 CONSTRUCTION OF A CONIC 247. Prop. X. If two points and two tangents of a conic he given, Uie chord of contact intersects the given chord in one of tuoo fixed points* . Let OP, OQ be the given tangents, A and B the given points, and G the intersection of AB and the chord of con- tact. Let OC be the polar of C, and let AB meet OC in D. Then (7 is on the polar of D, and therefore DBCA is an harmonic range. Also, O being on the polar of C, C'QGP is an harmonic range. Hence if two lines OG, OG' be found, which are har- monic with OA, OB, and also with OP, OQ, these lines intersect AB in two points G and D, through one of which the chord of contact must pass. Or thus, if the tangents meet AB in a and h, find the foci G and I) of the involution AB, ab; the chord of contact passes through one of these points. * I am indebted to Mr Wortbington for mach valoabls assistance in this chapter, and especially for the constmctions of Aitides 247, 249, 260, and 263. FBOH GIVEN CONDITIONS. 245 248. Prop. XI. Having given, three ■points and two tangents, to find the chord ofcoriact. In the preceding figure let OP, OQ be the tangents, and A, B, E the points. Find OG, 00' harmonic with OA, OB, and OP, OQ ; also find OF, OG harmonic with OA, OE and OP, OQ. Then any one of the four lines joining C or D to F or G is a chord of contact, and the chord of contact and points of contact being known, the case reduces to that of Art. 244. Hence four such conies can in general be described. 249. Prop. XII. To describe a conic, passing through two given points, and touching three given straight lines. Let AB, the line joining the given points, meet the given tangents QR, RP, PQ, in N, M, 1. Find the foci G, D of the involution A, B and L, M; Then YZ, the polar of P, passes through G or D, Art 247. Also find the foci, E, F, of the involution A, B, and M, N; then XT, the polar of R, passes through F or E. Let ZX meet PR in T; then T is on the polar of Q, and QFisthepolar of r. Hence TXC/i? is harmonic; therefore MEVG is harmonic. 246 coNSTBUcnoN of a conic This determines V, and, joining QV. -we obtain the point of contact F. Then, joining 7C and YE, Z and X are obtained, and X, T, Z being points of contact, we have five points, and can describe the conic by the construction of Art. 245, or by that of Art. 252. Since either G or D may be taken with B or F, there are in general four solutions of the problem. 250. Prop. XIII. To describe a conic, having given four points and one tangent. Let A, B, C, D be the given points, and complete the quadrilateral. 1 pk in a. constant ratio, and therefore 8 is the focus and kl the directrix of the section ap. 266. If the curve be a parabola focus 8', the proof is as follows : PT* = PG' - ECP = PN* + NGP'-EM*-MG' = MN (NG + MG) -^AS.MN = MN {NO + MG) - 2MG . MN = MN\ It will be found that the theorem is also true for an hyperboloid of two sheets, and for an hyperboloid of one sheet, but that in the latter case the constant ratio of PT to PL is not that of 8C to AC. 267. The geometrical enunciation of the theorem also requires modification in several cases. To illustrate the difficulty, take the paraboloid, and observe that if the normal coKoms. 263 at E cuts the axis in O, and if be the centre of curvature at^, A6>A0, and the radius of the circle is never less than AO. This shews that a circle the radius of which is less than AO cannot be drawn so as to touch the conic in two points. We may mention one exceptional case in which the theorem takes a simple form. In general EG' = EM* + MQ^ = 4^SP . PF, 80 that the associated conic is a parabola. If the conic is a circle, the associated conic is a rectangular hyperbola. If the conic is an ellipse, the axes of which are indefinitely small, that is, if it is reduced to a point, the associated conic lapses into two straight lines, which are at right angles to each other if the point is the limit of a circle. 278. If the point E be outside the conic, or, in other words, if the polar of E intersect the conic, it is not possible to project the conic into a circle, so that the projection of E shall be the centre of the circle. In this case the conic can be projected into a rectangular hyperbola, having the projection of the point E for its centre. Let RUhe the chord of contact of the tangents from E, and take any point on the surface of the sphere of which RUis& diameter. Then the projection of the conic from the vertex on any plane parallel to J20^will be an hyperbola, and, since ROUiB a right angle, it will be a rectangular hyperbola. 279. If two conies in a plame are entirely exterior to each other, they can in general be projected, from, the sa/me vertex, into circles on the sa/me plane. Draw four parallel tangents to the conies, and let ^be the point of intersection of the diameters, PGp and QOq, joining the points of contact. Also, let FR, FR' be the ordinates through F, parallel to the tangents, of the associated conies. If ^ is so situated that these ordinates are equal, the locus of the vertices from which the two conies can be pro- CONICAL PBOJECnON. 273 jected into circles will be the same, that is, it will be the circle of which F is the centre, and FR the length of the radius, in the plane through F perpendicular to FR. In this case, taking any point on the circle as the vertex, the two conies wUl be projected into circles on any plane parallel to the plane 0^22, and the centres of the circles will be the projections of E and E', the respective poles of FR with regard to the conies. 280. For different directions of the tangents, the points, F,R,R', will take up different positions, and for all directions of the tangents the loci of these points will be continuous curves. The loci of R and R' will, in general, intersect each other; that is to say, there will be, in general, positions of F such that FR and FR' are equal. Taking a particular case, let F be so situated that FR' is greater than FR ; then taking F at the point where its locus meets the conic 0, FR' vanishes, and therefore, between these two positions of F, there must be some position such that FR' is equal to FR. B. C. s. 18 274 CONICAL PROJECTION. We may observe that the locus of J" passes through G and G, the centres of the two conies. For, if CG is conjugate to the parallel tangents of the conic G, the point F is at G, and, if CG is conjugate to the parallel tangents of the conic G, the point F is at G. When FR' is equal to FR, the line thus obtained is called by Poncelet the Ideal Secant of the two conies. 281. In a similar manner if one conic is entirely inside another they can, in general, be projected into circles, one of which will be inside the other. Also two conies intersecting in two points may be pro- jected into two intersecting circles. Two conies intersecting in four points, or having contact at two points, cannot be projected into circles, but they can be projected into rectangular hyperbolas. 282. The method of projections enables us to extend to conies theorems which have been proved for a circle, and which involve, amongst other ideas, harmonic ranges, poles and polars, systems of collinear points, and systems of con- current lines. For instance, the theorems of Arts. 208 and 210 are easily proved for a circle, and by this method are at once extended to conies. Take as another instance Pascal's theorem, that the opposite sides of any hexagon inscribed in a conic intersect in three collinear points. If this be proved for a circle, the method of conical projection at once shews that it is true for any conic. The following very elementary proof of the theorem for a circle is given in Caicdcm's Th&trimes et Problimes de OSomitrie EUmentaire. Let ABGDEF be the hexagon, and let AB and ED meet in G, BG and FE in H, FA and DC in K. CONICAL PROJECTION. 275 Also let ED meet BC in M and AF in N, and let BC meet AFm L. Then we have the relations, LA.LF=LB.LG, MG.MB = MD.ME, NE.ND = NF.NA. K 18—2 276 CONICAL PBOJECnON. Also, the trian^e LMN being cut by the three tians- versals AG, DK, FN, we have the relations, LB.MG.NA = LA.MB.MG LC . MD.NK = LK.MC.ND LH.ME.NF=LF.MH.NE. Multiplying together these six equalities, taking account of the relations previously stated, and cutting out the factors common to the two products, we obtain LH.MG.FK=LK.MH.NG; .'. G, H, K are coUinear. Brianchon's theorem that, if a hexagon circumscribe a conic, the three opposite diagonals are concurrent is proved at once by observing that it is the 'reciprocal polar of Pascal's theorem. 283. Stereographic and Gnomonic Projections. If a point on the surface of a sphere be taken as the vertex of projection, and if the plane of projection be parallel to the tangent plane at the point, the projection of any figure drawn on the surface of the sphere is called its stereographic projection. If however the centre of 'the sphere be taken as the vertex of projection, and any plane be taken as the plane of projection, the projection of any figure drawn on the surface of the sphere is called its gnomonic projection. The stereographic projection of a circle drawn on the surface of the sphere is a circle ; for it can be easily shewn that it is a subcontrary section of the oblique cone formed by the vertex of projection and the circle on the sphere. The gnomonic projection of a circle on the sphere is obviously a conic. These projections are sometimes described in treatises on Astronomy, and in these treatises the vertex for stereographic projection is taken at the south pole of the earth, and, for gnomonic projection, at the centre of the earth ; and, in both COKICAL PBOJECTION. 277 cases, the plane of projection is taken parallel to the plane of the equator. 284. It win be seen that the discussions which are given in this chapter are confined entirely to cases of real projection. The chapter is intended to be simply an introduction to a large and important subject. The method of conical projections is due to Poncelet, and is worked out with great fulness and elaboration in his work entitled, TraiU des PropriiMs Projectives des Figures (Second edition, 1865, in two quarto volumes). In this work Poncelet extends the domain of pure geometry by the interpretation and use of the law of continuity, and, as one of its applications, by the introduction of the imaginary chord of intersection, or, as-it is called by Poncelet, the ideal secant of two conies. Amongst English writers, the student will find valuable chapters on projections in Salmons Conies, and in the large work on the Geometry of Conies, by Dr C. Taylor, the Master of St John's College, Cambridge. There is also an important work by Cremona, on Projective Geometry, which has been translated by Leudesdorf (Second edition, 1893). MISCELLANEOUS PROBLEMS. II. 1. If two conies have the same directrix, their common points are concyclic. 2. If a focal chord of a parabola is bisected in V and the line perpendicular to it through V meets the axis in G, SO is half the chord. 3. If the perpendicular to CP from a point P of an ellipse meets the auxiliary circle in Q, PQ varies as PJT. 4. A A' and Sff are the axes, and 8 is one of the foci of an elUpse ; if a parabola is described with oints, one in the given plane and the other -external to it, is constant, it will describe a conic, the section of a right cone whose vertex is the given external point. 44. In the construction of Art. 241 prove that CK' and CK are' respectively equal to the sum and difference of the semi-axes. 45. Given a tangent to an elUpse, its point of contact, and the director circle, construct the ellipse. 282 MISCELLANEOUS PROBLEMS. II. 46. If the tangent at any point P of an ellipse meet the auziliaiy circle in ^, R\ and if Q, A be the corresponding points on the eUipse, the tangents at Q and B. pass through tiie point P on the auxiliary circle corresponding to P. 47. In the ellipse PDPiy, FHCSPX and DCD are conjugate diameters ; CH is equal to CS, and the polw of S passes through a point X on i*"/" produced. If DX is drawn cutting the ellipse in ©, prove that HD is parallel to SQ. 48. If T is the pole of a chord of a conic, and F the intersection of the chord with the mrectrix, TSF is a right angle. 49. The x>olar of the middle point of a normal chord of a parabola meets the focal vector to the point of intersection of the chord with the directrix on the normal at the further end of the chord. 60. OP, OQ touch a parabola at P, Q; the tangent at R meets OP, OQ in S, r; if F is the intersection of PT, SQ, 0, R, F are collinear. 61. If from any point A a straight line AEK be drawn parallel to an asymptote of an hyperbola, and meeting the polar of ^ in iT and the curve in E, shew that AE=EK. 52. If a chord PQ of a parabola, whose pole is T, cut the directrix in F, the tangents from F bisect the angle PFT and its supplement. 63. A parabola, focus S, touches the three sides of a triangle ABC, bisecting the base BO in D ; prove that AS is a fourth proportional to- AD, AB, and AC. 54. A focal chord PSQ is drawn to a conic of which C is the centre; the tangents and normals at P and Q intersect in T and K respectively; shew that ST, SP, SK, SC form an harmonic penciL 65. POP is any diameter of an ellipse. The tangents at any two points D and E intersect in F. PE, PD intersect in Q. Shew tiiat FG is parallel to the diameter conjugate to POP". 56. A conic section is circumscribed by a quadrilateral ABCD : A is joined to the points of contact of CB, CD ; and C to the points of contact of AB, AD : prove that BD is a diagonal of the interior quadri- lateral thus formed. 67. A parabola touches the three lines CB, CA, AB in P, Q, R, and throv^h R a line parallel to the axis meets RQ in E; shew that ABEG is a parallelogram. 58. If a series of conies be inscribed in a given quadrilateral, shew that their centres lie on a fixed straight line. Shew also that this line passes through the middle points of the- diagonals. MISCEUJkNEOnS PROBLEMS. II. 283 69. Four points A, B, C, D are taken, no three of which lie in a straight line, and joined in every possible way ; and with another (Miint as focus four conies are described touching respectively the sides of the triangles BCD, CDA, DAB, ABC; prove that the four conies have a common tangent. 60. If the diagonals of a quadrilateral circumscribing a conic intersect in a focus, they are at right angles to one another, and the third diagonal is the corresponding directrix. 61. An ellipse and parabola have the same focus and directrix; tangents are drawn to the ellipse at the extremities of the major axis : shew that the diagonals of the quadrilateral formed by the four points where these tangents cut the parabola intersect in the common focus, and pass through the extremities of the minor axis of the ellipse. 62. Three chords of a circle pass through a point on the circum- ference ; with this point as focus and the chords as axes three parabolas are described whose parameters are inversely proportional to the chords; prove that the common tangents to the parabolas, taken two and two, meet in a point. 63. A circle is described touching the asymptotes of an hyperbola and having its centre at the focus. A tangent to thi% circle cuts the directrix in F. and has its pole with regard to the hyperbola at T. Prove that retouches the circle. 64. Two conies have a common focus : their corresponding direc- trices will intersect on their common chord, at a point whose focal distance is at right angles to that of the intersection of their common tangents. Also the parts into which either common tangent is divided by their common chord will subtend equal angles at the common focus. If the conies are parabolas, the inclination of their axes will be the angle subtended by the common tangent at the common focus. 65. The tangent at the point P of an hyperbola meets the directrix in Q; another point R is taken on the directrix such that QR subtends at the focus an angle equal to that between the transverse axis and an asymptote ; prove that the envelope of RP is a parabola. 66. If an hyperbola passes through the angular points of an equilateral triangle and has the centre of the circumscribing circle as focus, its eccentricity is the ratio of 4 to 3, and its latus rectum is one- third of the diameter of the circle. 67. 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. 68. Tangents are drawn at two points P, P" on an ellipse. If any tangent be drawn meeting those at P, P' in R, K, shew that the line bisecting the angle RSR intersects RR on a fixed tangent to the ellipse. 284 MISCELLANEOUS PROBLEMS. IL 69. The choids of a conic which subtend the same angle at the focus all touch another conic having the same focus and directrix. 70. Two conies have a common focus S and a common directrix, And tangents TP, TP' are drawn to one from any point on the other and meet the directrix in F and F'. Prove that the angles PSP, PSF are equal and constant. 71. A rectangular hyperbola circumscribes a triangle ABC; if D, E, F are the feet of the peipeudiculars from A, B, C on the opposite sides, the loci of the poles of the sides of the triangle ABC are the lines EF, FD, DE. 72. If two of the sides of a triangle, inscribed in a conic, pass through fixed points, the envelope of the third side is a conic. 73. If two circles be inscribed in a conic, and tangents be drawn to the circles from any point in the conic, the sum or ditiierence of these tangents is constant, according as the point does or docs not lie between the two chords of contact. 74. The four common tangents of two conies intersect two and two ' on the sides of the common s^-conjugate triangle of the conies. 75. Prove that a right cylinder, upon a given elliptic base, can be -cut in two ways so that the curve of section may be a circle; and that a sphere can always be drawn through any two circular sections of opposite systems. 76. An ellipse revolves about its major axis, and planes are drawn through a focus cutting the surface thus formed. Prove that the locus of the centres of the different sections is a surface formed by the revolution of an ellipse about CS where C or 8 are respectively the centre and focus of the original ellipse. 77. Given five tangents to a conic, find, by aid of Brianchon'a theorem, the points of contact 78. The alternate angular points of any pentagon ABCDE are joined, thus forming another pentagon whose corresponding angular points are a, b, e,d,e; Aa, Bb, Cc, Dd, Ee are joined and produced to meet the opposite sides of ABCDE in a, S, y,i,t; shew that if .^ be joined with the middle point of yS, B with the middle point of it. Sec., these five lines meet in a point. 79. If a conic be inscribed in a triangle, the lines joining the angular points to the points of contact of the opposite sides are con- current. 80. If a quadrilateral circumscribe a conic, the intersection of the lines joining SONS LONDON : YORK STREET, COVENT GARDEN NEW YORK: 66, FIFTH AVENUE; AND BOMBAY CAMBRIDGE : DEIGHTON, BELL & CO. October, 1894 CONTENTS. 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