Table Of ContentANALYTICAL GEOMETRY
2D and 3D
P. R. Vittal
Visiting Professor
Department of Statistics
University of Madras
Chennai
Chennai • Delhi
Brief Contents
About the Author
Preface
1 Coordinate Geometry
2 The Straight Line
3 Pair of Straight Lines
4 Circle
5 System of Circles
6 Parabola
7 Ellipse
8 Hyperbola
9 Polar Coordinates
10 Tracing of Curves
11 Three Dimension
12 Plane
13 Straight Line
14 Sphere
15 Cone
16 Cylinder
Contents
About the Author
Preface
1 Coordinate Geometry
1.1 Introduction
1.2 Section Formula
Illustrative Examples
Exercises
2 The Straight Line
2.1 Introduction
2.2 Slope of a Straight Line
2.3 Slope-intercept Form of a Straight Line
2.4 Intercept Form
2.5 Slope-point Form
2.6 Two Points Form
2.7 Normal Form
2.8 Parametric Form and Distance Form
2.9 Perpendicular Distance on a Straight Line
2.10 Intersection of Two Straight Lines
2.11 Concurrent Straight Lines
2.12 Angle between Two Straight Lines
2.13 Equations of Bisectors of the Angle between Two Lines
Illustrative Examples
Exercises
3 Pair of Straight Lines
3.1 Introduction
3.2 Homogeneous Equation of Second Degree in x and y
3.3 Angle between the Lines Represented by ax2 + 2hxy + by2 = 0
3.4 Equation for the Bisector of the Angles between the Lines Given by ax2 + 2hxy + by2 = 0
3.5 Condition for General Equation of a Second Degree Equation to Represent a Pair of Straight
Lines
Illustrative Examples
Exercises
4 Circle
4.1 Introduction
4.2 Equation of a Circle whose Centre is (h, k) and Radius r
4.3 Centre and Radius of a Circle Represented by the Equation x2 + y2 + 2gx + 2fy + c = 0
4.4 Length of Tangent from Point P(x1, y1) to the Circle x2 + y2 + 2gx + 2fy + c = 0
4.5 Equation of Tangent at (x1, y1) to the Circle x2 + y2 + 2gx + 2fy + c = 0
4.6 Equation of Circle with the Line Joining Points A (x1, y1) and B (x2, y2) as the ends of Diameter
4.7 Condition for the Straight Line y = mx + c to be a Tangent to the Circle x2 + y2 = a2
4.8 Equation of the Chord of Contact of Tangents from (x1, y1) to the Circle x2 + y2 + 2gx + 2fy + c
= 0
4.9 Two Tangents can Always be Drawn from a Given Point to a Circle and the Locus of the Point
of Intersection of Perpendicular Tangents is a Circle
4.10 Pole and Polar
4.11 Conjugate Lines
4.12 Equation of a Chord of Circle x2 + y2 + 2gx + 2fy + c = 0 in Terms of its Middle Point
4.13 Combined Equation of a Pair of Tangents from (x1, y1) to the Circle x2 + y2 + 2gx + 2fy + c =
0
4.14 Parametric Form of a Circle
Illustrative Examples
Exercises
5 System of Circles
5.1 Radical Axis of Two Circles
5.2 Orthogonal Circles
5.3 Coaxal System
5.4 Limiting Points
5.5 Examples (Radical Axis)
5.6 Examples (Limiting Points)
Exercises
6 Parabola
6.1 Introduction
6.2 General Equation of a Conic
6.3 Equation of a Parabola
6.4 Length of Latus Rectum
6.5 Different Forms of Parabola
Illustrative Examples Based on Focus Directrix Property
6.6 Condition for Tangency
6.7 Number of Tangents
6.8 Perpendicular Tangents
6.9 Equation of Tangent
6.10 Equation of Normal
6.11 Equation of Chord of Contact
6.12 Polar of a Point
6.13 Conjugate Lines
6.14 Pair of Tangents
6.15 Chord Interms of Mid-point
6.16 Parametric Representation
6.17 Chord Joining Two Points
6.18 Equations of Tangent and Normal
6.19 Point of Intersection of Tangents
6.20 Point of Intersection of Normals
6.21 Number of Normals from a Point
6.22 Intersection of a Parabola and a Circle
Illustrative Examples Based on Tangents and Normals
Illustrative Examples Based on Parameters
Exercises
7 Ellipse
7.1 Standard Equation
7.2 Standard Equation of an Ellipse
7.3 Focal Distance
7.4 Position of a Point
7.5 Auxiliary Circle
Illustrative Examples Based on Focus-directrix Property
7.6 Condition for Tangency
7.7 Director Circle of an Ellipse
7.8 Equation of the Tangent
7.9 Equation of Tangent and Normal
7.10 Equation to the Chord of Contact
7.11 Equation of the Polar
7.12 Condition for Conjugate Lines
Illustrative Examples Based on Tangents, Normals, Pole-polar and Chord
7.13 Eccentric Angle
7.14 Equation of the Chord Joining the Points
7.15 Equation of Tangent at ‘θ’ on the Ellipse
7.16 Conormal Points
7.17 Concyclic Points
7.18 Equation of a Chord in Terms of its Middle Point
7.19 Combined Equation of Pair of Tangents
7.20 Conjugate Diameters
7.21 Equi-conjugate Diameters
Illustrative Examples Based on Conjugate Diameters
Exercises
8 Hyperbola
8.1 Definition
8.2 Standard Equation
8.3 Important Property of Hyperbola
8.4 Equation of Hyperbola in Parametric Form
8.5 Rectangular Hyperbola
8.6 Conjugate Hyperbola
8.7 Asymptotes
8.8 Conjugate Diameters
8.9 Rectangular Hyperbola
Exercises
9 Polar Coordinates
9.1 Introduction
9.2 Definition of Polar Coordinates
9.3 Relation between Cartesian Coordinates and Polar Coordinates
9.4 Polar Equation of a Straight Line
9.5 Polar Equation of a Straight Line in Normal Form
9.6 Circle
9.7 Polar Equation of a Conic
Exercises
10 Tracing of Curves
10.1 General Equation of the Second Degree and Tracing of a Conic
10.2 Shift of Origin without Changing the Direction of Axes
10.3 Rotation of Axes without Changing the Origin
10.4 Removal of XY-term
10.5 Invariants
10.6 Conditions for the General Equation of the Second Degree to Represent a Conic
10.7 Centre of the Conic Given by the General Equation of the Second Degree
10.8 Equation of the Conic Referred to the Centre as Origin
10.9 Length and Position of the Axes of the Central Conic whose Equation is ax2 + 2hxy + by2 = 1
10.10 Axis and Vertex of the Parabola whose Equation is ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
Exercises
11 Three Dimension
11.1 Rectangular Coordinate Axes
11.2 Formula for Distance between Two Points
11.3 Centroid of Triangle
11.4 Centroid of Tetrahedron
11.5 Direction Cosines
Illustrative Examples
Exercises
12 Plane
12.1 Introduction
12.2 General Equation of a Plane
12.3 General Equation of a Plane Passing Through a Given Point
12.4 Equation of a Plane in Intercept Form
12.5 Equation of a Plane in Normal Form
12.6 Angle between Two Planes
12.7 Perpendicular Distance from a Point on a Plane
12.8 Plane Passing Through Three Given Points
12.9 To Find the Ratio in which the Plane Joining the Points (x1, y1, z1) and (x2, y2, z2) is Divided
by the Plane ax + by + cz + d = 0.
12.10 Plane Passing Through the Intersection of Two Given Planes
12.11 Equation of the Planes which Bisect the Angle between Two Given Planes
12.12 Condition for the Homogenous Equation of the Second Degree to Represent a Pair of Planes
Illustrative Examples
Exercises
13 Straight Line
13.1 Introduction
13.2 Equation of a Straight Line in Symmetrical Form
13.3 Equations of a Straight Line Passing Through the Two Given Points
13.4 Equations of a Straight Line Determined by a Pair of Planes in Symmetrical Form
13.5 Angle between a Plane and a Line
13.6 Condition for a Line to be Parallel to a Plane
13.7 Conditions for a Line to Lie on a Plane
13.8 To Find the Length of the Perpendicular from a Given Point on a Line
13.9 Coplanar Lines
13.10 Skew Lines
13.11 Equations of Two Non-intersecting Lines
13.12 Intersection of Three Planes
13.13 Conditions for Three Given Planes to Form a Triangular Prism
Illustrative Examples
Illustrative Examples (Coplanar Lines and Shortest Distance)
Exercises
14 Sphere
14.1 Definition of Sphere
14.2 The equation of a sphere with centre at (a, b, c) and radius r
14.3 Equation of the Sphere on the Line Joining the Points (x1, y1, z1) and (x2, y2, z2) as Diameter
14.4 Length of the Tangent from P(x1, y1, z1) to the Sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0
14.5 Equation of the Tangent Plane at (x1, y1, z1) to the Sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d
= 0
14.6 Section of a Sphere by a Plane
14.7 Equation of a Circle
14.8 Intersection of Two Spheres
14.9 Equation of a Sphere Passing Through a Given Circle
14.10 Condition for Orthogonality of Two Spheres
14.11 Radical Plane
14.12 Coaxal System
Illustrative Examples
Exercises
15 Cone
15.1 Definition of Cone
15.2 Equation of a Cone with a Given Vertex and a Given Guiding Curve
15.3 Equation of a Cone with its Vertex at the Origin
15.4 Condition for the General Equation of the Second Degree to Represent a Cone
15.5 Right Circular Cone
15.6 Tangent Plane
15.7 Reciprocal Cone
Exercises
16 Cylinder
16.1 Definition
16.2 Equation of a Cylinder with a Given Generator and a Given Guiding Curve
16.3 Enveloping Cylinder
16.4 Right Circular Cylinder
Illustrative Examples
Exercises
About the Author
P. R. Vittal was a postgraduate professor of Mathematics at Ramakrishna
Mission Vivekananda College, Chennai, from where he retired as Principal in
1996. He was a visiting professor at Western Carolina University, USA, and has
visited a number of universities in the USA and Canada in connection with his
research work. He is, at present, a visiting professor at the Department of
Statistics, University of Madras; Institute of Chartered Accountants of India,
Chennai; The Institute of Technology and Management, Chennai; and National
Management School, Chennai, besides being a research guide in Management
Science at BITS, Ranchi.
Professor Vittal has published 30 research papers in journals of national and
international repute and guided a number of students to their M.Phil. and Ph.D.
degrees. A fellow of Tamil Nadu Academy of Sciences, his research topics are
probability, stochastic processes, operations research, differential equations and
supply chain management. He has authored about 30 books in mathematics,
statistics and operations research.
To my grandchildren
Aarav and Advay
Preface
A successful course in analytical geometry must provide a foundation for future
work in mathematics. Our teaching responsibilities are to instil certain technical
competence in our students in this discipline of mathematics. A good textbook,
as with a good teacher, should accomplish these aims. In this book, you will find
a crisp, mathematically precise presentation that will allow you to easily
understand and grasp the contents.
This book contains both two-dimensional and three-dimensional analytical
geometry. In some of the fundamental results, vector treatment is also given and
therefrom the scalar form of the results has been deduced.
The first 10 chapters deal with two-dimensional analytical geometry. In
Chapter 1, all basic results are introduced. The concept of locus is well
explained. Using this idea, in Chapter 2, different forms for the equation of a
straight line are obtained; all the characteristics of a straight line are also
discussed. Chapter 3 deals with the equation of a pair of straight lines and its
properties. In Chapters 4 and 5, circle and system of circles, including coaxial
system and limiting points of a coaxial system, are analysed.
Chapters 6, 7 and 8 deal with the conic sections—parabola, ellipse and
hyperbola. Apart from their properties such as focus and directrix, their
parametric equations are also explained. Special properties such as conormal
points of all conics are described in details. Conjugate diameters in ellipse and
hyperbola and asymptotes of a hyperbola and rectangular hyperbola are also
analysed with a number of examples. A general treatment of conics and tracing
of conics is also provided.
In Chapter 9, we describe polar coordinates, which are used to measure
distances for some special purposes. Chapter 10 examines the conditions for the
general equation of the second degree to represent the different types of conics.
