Analytical Geometry: Two Dimensions

317 Pages·2022·8.29 MB·english

by  Sibdas Karmakar

Analytical Geometry:
Two Dimensions
Prof. Sibdas Karmakar
Retd. Associate Professor of Mathematics
Ramananda College, Bishnupur, Bankura
Dr. Samiran Karmakar
Assistant Professor of Mathematics
Bankura Sammilani College, Bankura
Levant Books
India
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ISBN:9781032270203(hbk)
ISBN:9781003293248(ebk)
DOI:10.4324/9781003293248
This book is dedicated to PARNIKA KARMAKAR, the youngest
daughter of Samiran.

Preface
Thisbookentitled”Analytical Geometry: Two Dimensions”hasbeen
written in accordance with the modern trend in the teaching of Geometry.
Inpreparingthisvolume, theauthorshaveendeavouredtowriteadrillbook
for beginners which presents the fundamental concepts of this topic. The
number of propositions with formal proofs has been reduced and stress laid
on the solutions of the problems. Authors believe that this text is complete
for students finishing their studies of Mathematics with a course in Analytic
Geometry of Two Dimensions. All new materials as well as the old one have
been treated from the same standpoint.
Thepresentworkdoesnotclaimtobeoriginal. Author’saimofthisbook
is to make the subject thoroughly intelligible. All the important properties
of the conics have been discussed either in the articles or in illustrative
examples. A large number of problems have been completely solved and
exercises containing carefully graded and motivating examples have been
incorporated at the end of each chapter.
Highlights of some specialties:
The book is self-contained.
•
Simple, comprehensive, lucid and rigorous discussions are provided.
•
• Basic school level algebra and geometry is sufficient to understand the
topics of this book.
• Treatment of matrix theory has been shown in some topics, e.g., clas-
sification of conics, of this book.
Attentioniscalledtothemethodoftreatment. Thesubjectisdeveloped
after the Euclidean method of definition and theorems. However, the formal
presentation is avoided. Emphasis has been given on the analytical side
everywhere.
v
The authors hope the book will prove useful not only to the students
of Mathematics but also to the users of other branches of science and tech-
nology as well as in the applications of recently developing Business and
financial sectors throughout the world.
In bringing out this book, Authors would wish to record their appreci-
ation to the Publishers and Printers for their care and efforts. Criticisms
and any constructive suggestions for the improvement of this book will be
gratefully acknowledged.
We are thankful to the students who has constantly inspired us to com-
plete this work by expressing the difficulties and problems they face fre-
quently in this subject.
Authors
vi
Contents
Preface v
0 Preliminaries 1
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Terminologies and Fundamental Concepts . . . . . . . . . . . 1
0.2.1 Directed Lines . . . . . . . . . . . . . . . . . . . . . . 1
0.2.2 Directed Line Segment and its Value . . . . . . . . . . 1
0.2.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2.4 Half Line . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.3 Some Tips and Tricks . . . . . . . . . . . . . . . . . . . . . . 3
0.3.1 On Coordinate and Locus . . . . . . . . . . . . . . . . 3
0.3.2 Straight Lines . . . . . . . . . . . . . . . . . . . . . . . 9
0.3.3 The Circle . . . . . . . . . . . . . . . . . . . . . . . . . 12
0.3.4 The Parabola . . . . . . . . . . . . . . . . . . . . . . . 14
0.3.5 The Ellipse . . . . . . . . . . . . . . . . . . . . . . . . 15
0.3.6 The Hyperbola . . . . . . . . . . . . . . . . . . . . . . 16
0.3.7 Common Properties of all Conics . . . . . . . . . . . . 18
0.3.8 Properties of Tangents and Normals on Conics . . . . 18
0.3.9 Conjugate Diameters . . . . . . . . . . . . . . . . . . . 19
1 Transformation of Rectangular Cartesian Coordinates 21
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2 Coordinate Transformation Formulas . . . . . . . . . . . . . . 21
1.2.1 Translation (or transformation by translation of axes) 21
1.2.2 Rotation (or transformation by rotation of axes) . . . 22
1.2.3 Rigid Motion or Orthogonal Transformation . . . . . . 24
1.3 Use of Matrix Notation . . . . . . . . . . . . . . . . . . . . . 24
1.3.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.2 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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1.3.3 Rigid Motion . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Definition of Orthogonal Transformation . . . . . . . . . . . . 26
1.5 General Orthogonal Transformation . . . . . . . . . . . . . . 26
1.6 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.7 Purpose of Transformation of Axes . . . . . . . . . . . . . . . 27
1.8 To Investigate whether there is any Fixed Point under Trans-
lation, Rotation or Rigid Motion . . . . . . . . . . . . . . . . 27
1.8.1 Fixed Point under Translation . . . . . . . . . . . . . 27
1.8.2 Fixed Point under Rotation . . . . . . . . . . . . . . . 28
1.8.3 Fixed Point under Rigid Motion . . . . . . . . . . . . 28
1.9 Invariants of Translation, Rotation and Rigid Motions . . . . 29
1.10 Work Out Examples . . . . . . . . . . . . . . . . . . . . . . . 34
1.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2 General Equation of Second Degree: Classification of Con-
ics 55
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.1 Centre of a second order curve . . . . . . . . . . . . . 56
2.3 Determination of Centre of a Second Order Curve . . . . . . 57
2.4 Solution of Simultaneous Equations Using Matrix Notation . 60
2.5 Canonical Form of an Equation of a Second Degree Curve . . 61
2.6 Reduction of an Equation of a Conic to Canonical Form . . . 65
2.7 Rank and Classification of Second Order Curves . . . . . . . 69
2.8 Conic through Common Points . . . . . . . . . . . . . . . . . 69
2.9 Conditions for Pair of Straight Lines and proper Conics . . . 70
2.10 Worked Out Examples . . . . . . . . . . . . . . . . . . . . . . 74
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3 Polar Coordinates and Equations 101
3.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Discussion on Polar Coordinates . . . . . . . . . . . . . . . . 101
3.3 Relation between Polar and Cartesian Coordinates of a Point 102
3.4 Distance between Two Points . . . . . . . . . . . . . . . . . . 103
3.5 Area of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.6 Straight Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.6.1 Equationofastraightlineinpolarcoordinatesingen-
eral form . . . . . . . . . . . . . . . . . . . . . . . . . 104
viii
3.6.2 Equation of a Line Passing through the Two Given
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.6.3 Polar equation of a line in normal form . . . . . . . . 107
3.7 The Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.7.1 Polar equation of a circle . . . . . . . . . . . . . . . . 108
3.8 Polar Equation of a Conic . . . . . . . . . . . . . . . . . . . . 111
3.8.1 Definition of a conic . . . . . . . . . . . . . . . . . . . 111
3.8.2 Polar equation of a conic whose semi latus rectum is
l and focus is the pole . . . . . . . . . . . . . . . . . . 111
3.9 Polar Equation of the Chord of a Given Conic . . . . . . . . . 114
3.10 Polar Equation of the Tangent to a Given Conic . . . . . . . 115
3.11 Polar Equation of the Normal to a Given Conic . . . . . . . . 116
3.12 Chord of Contact . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.12.1 Equation of Chord of Contact . . . . . . . . . . . . . . 117
3.13 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.13.1 Equation of the asymptote . . . . . . . . . . . . . . . 120
3.14 Polar Equation of the Directrices of an Ellipse . . . . . . . . . 121
3.15 A Few Properties . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.16 Worked-Out Examples . . . . . . . . . . . . . . . . . . . . . . 124
3.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4 Pair of Straight Lines 165
4.1 General Equations . . . . . . . . . . . . . . . . . . . . . . . . 165
4.2 To Find a Necessary Condition that the General Equation of
Second Degree Should Represent a Pair of Straight Lines . . 167
4.3 Angle Between a Pair of Lines . . . . . . . . . . . . . . . . . . 174
4.4 Equation of Bisectors of the Angle between a pair of straight
lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.5 Equation of Two Lines Joining the Origin to the Points in
Which a Line Meets a Conic . . . . . . . . . . . . . . . . . . . 176
4.6 Worked Out Examples . . . . . . . . . . . . . . . . . . . . . . 177
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5 Tangents and Normals, Pair of Tangents, Chord of Contact
& Pole and Polar 209
5.1 Tangent to a Curve . . . . . . . . . . . . . . . . . . . . . . . . 209
5.1.1 To Find the Equation of the Tangent to a Conic at a
Given Point on the Conic . . . . . . . . . . . . . . . . 209
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