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An Elementary Treatise on Conic Sections

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An Elementary Treatise on Conic Sections :# a-r.-.-'. :t:-x ^^i^-a '!'•'"/ !.'; The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924063287340 CORNELL UNIVERSITY LIBRARY BY CHARLES SMITH, M.A. ELEMENTARY ALGEBRA. Eleventh Impression. Revised and enlarged. Globe 8vo. 4J. 64. Key. lOi-. 6d. For the Second Edition the whole book was thoroughly revised, and the early chapters remodelled and simplified. The number of examples was also very greatly increased, A Chapter on Graphs has been added. A TREATISE ON ALGEBRA. Eighth Impression. Crown 8vo. Is. 6d. Key. los. 6d. This book has been made more complete by the addition of a Chapter on the Theory of Equations. AN ELEMENTARY TREATISE ON SOLID GEOMETRY. Ninth Impression. Crown 8vo. gj. 6d. GEOMETRICAL CONICS. Third Impression. Crown 8vo. 6s. AN ELEMENTARY TREATISE ON CONIC SECTIONS. Eighteenth Impression. Crown 8vo. 'js. 6d. Key. ioj. td. By C. Smith, M.A., and S. Bryant, D.Sc. EUCLID'S ELEMENTS OF GEOMETRY. Books I —VI. and XI. Globe 8vo. 4^. 6d. Book I. is. Books I. and II. ij. 6d. Books III. and IV. is. Books I. —IV. is. Books VI. and XI. \s. 6d. Key. 8j. 6d. MACMILLAN AND CO. Ltd., LONDON. CONIC SECTIONS. AN ELEMENTARY TREATISE ON CONIC SECTIONS CHARLES SMITH, M.A., MASTEB OF SIDNEY SUSSEX COLLEGE, CAMBRIDaE. Honlion MACMILLAN AND CO., Limited NE"W YOEK: THE MACMILLAN COMPANY 1904 All rights reserved. First Edition printed i88». Second Edition printed 18S3. Reprinted with slight corrections 1884, with slight ccrr- rections 1885, 1887, 1888, 1889, January and November 169O1 1892, 1893, 1894, 1896, 1898, 1899, I9°i> '9°'< 1904 PKEFACE TO THE FIEST EDITION. In the foUowiDg work I have investigated the more elementary properties of the Ellipse, Parabola, and Hy- perbola, defined with reference to a focus and directrix, before considering the General Equation of the Second Degree. I believe that this arrangement is the best for beginners. The examples in the body of each chapter are for the most part very easy applications of the book-work, and have been carefully selected and arranged to illustrate the principles of the subject. The examples at the end of each chapter are more difficult, and include very many of those which have been set in the recent University and College examinations, and in the examinations for Open Scholarships, in Cambridge. The answers to the examples, together with occasional hints and solutions, are given in an appendix. I have also, in the body of the work, given complete solutions of some illustrative examples, which I hope will be found especially useful. s. c. s. b ; VI PREFACE. Although I have endeavoured to present the ele- mentary parts of the subject in as simple a manner as possible for the benefit of beginners, I have tried to make the work in some degree complete; and have therefore included a chapter on Trilinear Co-ordinates, and short accounts of the methods of Reciprocation and Conical Pro- jection. For fuller information on these latter subjects the student should consult the works of Dr Salmon, Dr Ferrers, and Dr C. Taylor, to all of whom it will be seen that I am largely indebted. I am indebted to several of my friends for their kind- ness in looking over the proof sheets, for help in the verification of the examples, and for valuable suggestions and it is hoped that few mistakes have escaped detection. CHARLES SMITH. Sidney Sussex Collegi, April, 1882. PREFACE TO THE SECOND EDITION. The second edition has been carefully revised,, and some additions have been made, particularly in the last Chapter. Sidney Sussex College. July, 1883. CONTENTS. [The Articles marked with'an asterisk may be omitted by beginners until after they have read Chapter /A'.] PAGE CHAPTER I. CO-OEDINATES .... 1 CHAPTEE It The Sibaight Line . 12 Examples on Chapter II . 43 CHAPTEE in. Change of Axes. Anhakmonio Eatios or Ckoss Eatios. Involution 47 CHAPTEE IV. The Cikcle . ., . .60 Examples on Chapter IV . 83 CHAPTEE V. The Pakabola , 88 Examples on Chapter V 105 CHAPTEE VI. The Ellipse . 112 Examples on Chapter VI . 138 CHAPTEE Vn. The Htpebbola . 146 Examples on Chapter VII ... ... 162 CHAPTEE VIH. Polab Equation of a Conic, the focus being the pole 166 Examples on Chapter VIII 176 CHAPTEE IX. Genekal Equation of ihe^eoond Degeee. Every curve whose equation is of the second degree is a conic 179 Co-ordinates of the centre of a conic 181 The Discriminant 182 Position and magnitude of the axes of a central conic . 183 vau CONTENTS. PAQE Axis and latus rectum of a parabola 184 Tracing eonios ... 185 Equation of the asymptotes of a conic . 188 Condition for a rectangular hyperbola . 189 Examples on Chapter IX . , 190 CHAPTEE X. Miscellaneous Pbopositions. Equation of the tangent at any point of a conic . 192 Condition that a given straight Une may touch a conic . 193 Equation of the polar of any point with respect to a conic . 194 Conjugate points and conjugate lines 195 A chord o^ a conic is out harmonically by a point and its polar 195 Diameters of a conic 196 Condition that two given lines may be parallel to conjugate diameters 196 Equi-conjugate diameters of a conic 197 Common conjugate diameters of two conies . 197 Segments of chords of a conic 198 Pairs of common chords of a circle and a conic are equally inclined to the axes of the conic 199 Meaning of S - XS'= 0, S - tei; = and S - \u2 = . 200 Equation of a pair of tangents 202 Equation of the director-circle ... 203 Four foci of a conic ... ... 204 Eccentricities of a conic . 205 Equations giving the foci . ... 207 Equation of the axes ... .... 209 Equation of a conic referred to tangent and normal . 211 Normals 211 Similar conies 215 Examples on Chapter X . .... 219 CHAPTEE XI. Systems of Conics. One conic through five points 231 Conics through four points. [See also 295] . 233 Two parabolas through four points , 234 Centre-locus of oonics through four points. [See also 296] 234 Diitgonal-points of a quadrangle are angular points of a triangle self-polar with respect to any circumscribing conic. [See also 297.] 237 CONTENTS. IX PAQE Diagonals of a quadrilateral are sides of a triangle self-polar with respect to any inscribed conic .... 238 Centre-loous of conies touching four fixed lines. [See also 137, 270, and 297] 240 Parabola touching the axes of co-ordinates .... 241 Confocal conies 244 Osculating conies . 251 Pairs of common chords of two conies . 253 Invariants ... ... 254 Examples on Chapter XI. ... 257 CHAPTER XII. Envelopes. Envelopes, how found .... 264 Tangential co-ordinates and equations 267 Director-circle of envelope (cf. 203) . 268 Foci of envelope . 269 Meaning of the tangential equation iS-XS'=0 . 269 Director-circles of conies which touch four given straight lines have a common radical axis. [See also 308 and 319.] . 270 Examples . 271 CHAPTER Xni. Tbilineab Co-obmnaies. Definition of Trilinear Co-ordinates . 276 Equations and intersections of straight lines . 278 The hne at infinity . 280 Parallel lines .280 Condition of perpendicularity 281 Length of perpendicular from a point on a line . 282 Co-ordinates of four points in the form ± /, ± j, ± 7t . 283 Equation of four lines in the form Za±m/3±«7=0 . 284 Examples ... 285 Tangent ..... 286 Condition of tangenoy .... 287 Co-ordinates of the centre of a conic . 288 Condition for a parabola . 288 The asymptotes 289 Condition for a rectangular hyperbola 290 The circumscribing circle . 290 The circular points at infinity . 291 Conditions for a circle . 291 X CONTENTS. FAQE 292 Condition for ellipse, parabola, or hyperbola . Fooi 293 293 Equation of a circumscribing conic . • 294 Equation of an inscribed conic . • • Equation of the inscribed circle . 294 • 295 Conies through four fixed points . 296 Conies touching four fixed lines . Conies with respect to which the triangle of reference is self-polar 297 Conies referred to two tangents and the chord of contact . 298 The circle with respect to which the triangle of reference is . 298 self-polar ... The nine-point circle . 299 Pascal's theorem. [See also 328.] . 299 Brianohou's theorem . 301 Areal Co-obdinates . 302 Tangential Co-ordinates 303 Identical relation between the tangential co-ordinates of any straight line 304 Tangential equation of a conic . 305 The equation of the pole of a line 305 Examples on Chapter XIII. 309 CHAPTEB XIV. Eeoipbocal Polabs. Projections. Definition of polar reciprocal 315 The degree of a curve is the same as the class of its reciprocal 315 Examples of reciprocal theorems 317 Beeiprocation with respect to a circle 318 Eeciprocal of a circle with respect to a point . 320 Co-axial circles reciprocated Inio confocal conies . 322 Prcgection. Definition of projection 323 The projection of any curve is a curve of the same degree . 323 Projections of tangents, poles and polars .... 324 Straight lines meeting in a point projected into parallel straight lines 324 Any line can be projected to infinity, and at the same time any two angles into given angles 324 A quadrilateral projected into a' square 326 Any conic projected into a circle . 337 CONTENTS. XI FAQE A eystem of conies inscribed in a quadrilateral projected into eonfocal conies 327 Cross ratios of pencils and ranges unaltered by projection . 329 Conies through four fixed points are cut by any straight line in pairs of points in involution 329 The cross ratio of a pencil of four lines equal to that of the range formed by their poles . 329 Anharmonic j)roperties of points on a conic, and of tangents to a eanio ..
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