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THE ELEMENTS OF COORDINATE GEOMETRY With Solutions of Examples S. L. LONEY Compiled by S. Pal M H AT VALLEY THE ELEMENTS OF COORDINATE GEOMETRY With Solutions of Examples BY S. L. LONEY, M.A. LATE FELLOW OF SIDNEY SUSSEX COLLEGE, PROFESSOR AT THE ROYAL HOLLOWAY COLLEGE M H AT VALLEY Amazon Kindle Edition 2019 Published by Math Valley Publishing Copyright © 2019 by MathValley Compiled by S. Pal M H AT VALLEY Math Valley Publishing Brief Contents Preface iv Publisher's Note v About the Book vi Detailed Contents vii Chapter 1 Introduction: Some Algebraic Results 1-7 Chapter 2 Coordinates: Lengths of Straight Lines and· 8-30 Areas of Triangles Chapter 3 Locus: Equation to a Locus 31-39 Chapter 4 The Straight Line: Rectangular Coordinates 40-96 Chapter 5 The Straight Line. Polar Equations and Oblique 97-139 Coordinates. Miscellaneous Problems. Loci Chapter 6 On Equations Representing Two or More 140-180 Straight Lines Chapter 7 Transformation of Coordinates 181-193 Chapter 8 The Circle 194-265 Chapter 9 Systems of Circles 266-286 CONIC SECTIONS Chapter 10 The Parabola 287-344 Chapter 11 The Parabola (Continued) 345-383 Chapter 12 The Ellipse 384-459 Chapter 13 The Hyperbola 460-518 Chapter 14 Polar Equation to a Conic 519-543 Chapter 15 General Equation Tracing of Curves 544-590 Chapter 16 The General Conic 591-652 Chapter 17 Miscellaneous Propositions 653-716 About the Author 717 Preface In following work I have tried to present the elements of Coordinate Geometry in a manner suitable for Beginners and Junior Students. The present book deals only with Cartesian and Polar Coordinates. Within these limits I venture to hope that t he book is fairly complete, and that no propositions of very great importance have been omitted. The Straight Line and Circle have been treated more fully than the other portions of the subject, since it is generally in the elementary conceptions that beginners find great difficulties. There are a large number of Examples, over 1100 in all, and they are, in general, of an elementary character. The examples are especially numerous in the earlier parts of the book. I am much indebted to several friends for reading portions of the proof sheets, but especially to Mr . W. J. Dobbs, M.A. who has kindly read the whole of the book and made many valuable suggestions. For any criticisms, suggestions, or corrections, I shall be grateful. S. L. LONEY ROYAL HOLLOWAY COLLEGE FOR WOMEN, EGHAM, SURREY July 12, 1895. iv v Publisher’s Note The latest edition of the book ‘ Elements of Coordinate Geometry ’ by S. L. Loney has passed more than a century. Still now, this is a world wide accepted book in Mathematics and Mathematical Sciences. Digital copy of the book is available in the internet or in digital library. This is also a digital version of the book but arranged in different order for the reader who is directly reading the book using modern electronic devices like computer, digital reader, i-pad, i-phone, mobile, etc . In this context, a little modification is made in the organization of the book – answers and solution/hints of the practice e xamples at the end of an article is given immediately after those e xamples. This reduces the movement of digital pages of a thick book while also helping in the verification of the answer obtained by the reader. Also the reader do es not have to remember page number for this purpose. It saves both time and effort. A brief and detailed contents of the book are also given at the beginning of the book. Th is helps reader to search a topic as per requirement. PUBLISHER February, 2019 vi About the Book The book ‘Elements of Coordinate Geometry ’ by S. L. Loney is a fundamental book on analytical geometry in Mathematics for school and college students. Even after a century, it is still now an unparalleled book on two-dimensional coordinate geometry and it creates the backbone of a student who love s mathematics. This book consists of 17 chapters. Some fundamental algebraic results are stated in Chapter 1. Chapter 2 and Chapter 3 deal with the fundamentals of coordinate geometry. Chapters 4-6 deal with straight line and pair of straight lines. Chapter 7 state s the concept s of transformation of coordinates. Circle and system of circles are discussed in chapters 8 and 9. Conic sections that include parabola, ellipse and hyperbola are discussed in chapters 10 through 13. Discussions in chapters 1 -13 are based on rectangular a nd oblique Cartesian Coordinate system. Conics in polar coordinate system is discussed in Chapter 14. General equations are discussed in chapter s 15-16. Last chapter, Chapter 17 deals with miscellaneous propositions. The book ‘Elements of Coordinate Geomet ry’ by S. L. Loney is the father of all coordinate geometry books available in the present academic market. One should read this book carefully – it will create a mind with clear concepts and in depth knowledge on coordinate geometry. This helps the reader to solve any kind of problems on coordinate geometry of 2D. vii Detailed Contents Preface iv Publisher's Note v About the Book vi Detailed Contents vii-xv Chapter 1. INTRODUCTION: SOME ALGEBRAIC 1-7 RESULTS Quadratic equations 1 Determinant notation 2 Elimination 5 Chapter 2. COORDINATES: LENGTHS OF STRAIGHT 8-30 LINES AND· AREAS OF TRIANGLES Coordinates 8 EXAMPLES I 14 ANSWERS 16 SOLUTIONS/HINTS 16 EXAMPLES II 22 ANSWERS 23 SOLUTIONS/HINTS 23 Polar coordinates 25 EXAMPLES III 28 ANSWERS 29 SOLUTIONS/HINTS 30 Chapter 3. LOCUS: EQUATION TO A LOCUS 31-39 Equation to a curve 35 EXAMPLES IV 37 ANSWERS 38 SOLUTIONS/HINTS 38 viii Chapter 4. THE STRAIGHT LINE: RECTANGULAR 40-96 COORDINATES Equation of straight line: 40 Parallel to coordinate axis 40 ymxc 41 x y 42  1 a b xcos ysin p 44 AxByC 0 45 Straight line at infinity 47 Straight line through two points 48 EXAMPLES V 50 ANSWERS 51 SOLUTIONS/HINTS 52 Angle between straight lines 56 Conditions that they may be parallel and 57 perpendicular EXAMPLES VI 62 ANSWERS 63 SOLUTIONS/HINTS 64 Lengths of perpendiculars 70 EXAMPLES VII 72 ANSWERS 73 SOLUTIONS/HINTS 73 Point of intersection 75 Condition that three lines may meet in a point 76 Straight line which passes through the intersection of the 77 two straight lines Bisectors of angles between straight lines 79 Length of the straight line drawn through a given 81 point in a given direction to meet a given straight line EXAMPLES VIII 82 ANSWERS 86 SOLUTIONS/HINTS 87 Chapter 5. THE STRAIGHT LINE: POLAR EQUATIONS 97-139 AND OBLIQUE COORDINATES. MISCELLANEOUS PROBLEMS. LOCI Straight line in polar coordinates 97 Oblique coordinates 98 ix EXAMPLES IX 103 ANSWERS 104 SOLUTIONS/HINTS 104 Straight lines passing through fixed points 109 EXAMPLES X 114 ANSWERS 117 SOLUTIONS/HINTS 117 EXAMPLES XI 129 ANSWERS 131 SOLUTIONS/HINTS 132 Chapter 6. ON EQUATIONS REPRESENTING TWO OR 140-180 MORE STRAIGHT LINES Angle between two lines given by one equation 142 EXAMPLES XII 146 ANSWERS 146 SOLUTIONS/HINTS 146 General equation of the second degree 150 EXAMPLES XIII 154 ANSWERS 155 SOLUTIONS/HINTS 155 Homogeneous equation of the nth degree 159 EXAMPLES XIV 165 ANSWERS 169 SOLUTIONS/HINTS 169 Chapter 7. TRANSFORMATION OF COORDINATES 181-193 Alter the origin 181 Change the direction of axes of coordinates 182 EXAMPLES XV 184 ANSWERS 184 SOLUTIONS/HINTS 185 Degree of an equation 188 EXAMPLES XVI 191 ANSWERS 192 SOLUTIONS/HINTS 192 Chapter 8. THE CIRCLE 194-265 Definition 194 Equation of circle 194