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A Point on Coordinate Geometry

Shiv Dayal

Copyright © 2026, present Shiv S. Dayal

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”.

Dedication

Dedicated to my parents

Table of Contents

Preface
I.
1. Coordinates
1.1. Number Line
1.2. Coordinates
1.3. Cartesian Coordinate System
1.4. Distance Between Two Points
1.5. Section Formula
1.6. Area of a Trapezium
1.7. Area of a Triangle
1.7.1. Sign of Area of a Triangle
1.7.2. Condition of Collinearity of Three Points
1.8. Centroid of a Triangle
1.9. Incenter of a Triangle
1.10. Area of a Quadrilateral
1.11. Polar Coordinates
1.12. Polar Coordinates and Cartesian Coordinates
1.13. Problems
2. Locus
2.1. Finding Locus of a Point
2.2. Equation of Locus in Different Forms
2.3. Problems
3. Straight Lines
3.1. Angle of Inclination of a Line
3.2. Slope or Gradient of a Line
3.3. Equation of a Straight Line
3.4. Equation of Lines Parallel to Axes
3.4.1. Equation of x -axis
3.4.2. Equation of y -axis
3.4.3. Equations of Lines Parallel to y -axis
3.4.4. Equations of Lines Parallel to y -axis
3.5. Forms of Straight Line
3.5.1. Slope-Intercept Form
3.5.2. Point-Slope Form
3.5.3. Two-Point Form
3.5.4. Intercept Form
3.5.5. Normal Form
3.5.6. Distance Form or Parametric Form
3.5.7. General Euqation of a Straight Line
3.6. Representing General Equation in Standard Forms
3.6.1. Slope Intercept Form
3.6.2. Intercept Form
3.6.3. Normal Form
3.7. Angle between Two Straight Lines
3.7.1. Parallelism and Perpendicularity
3.7.2. To find the angle between straight lines a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0
3.7.3. Lines Parallel to Another Line
3.7.4. Lines Perpendicular to Another Line
3.8. Point of Intersection
3.8.1. Line Passing through Point of Intersection of Two Lines
3.9. Concurrency of Three Straight Lines
3.10. Two Sides of a Straight Line
3.11. Length of a Perpendicular
3.12. Bisectors of Angles between Straight Lines
3.12.1. Finding Bisector of the Acute and Obtuse Angles
3.12.2. Bisectors Between Lines Containing a Given Point
3.13. Problems
4. Pair of Straight Lines
4.1. Homogeneous Equations
4.2. Pair of Straight Lines Through Origin
4.3. Angle Between the Lines Represented by Second Degree Equation
4.4. Bisectors of Angles Between the Pair of Straight Lines
4.5. Condition for General Equation of Second Degree to Represent a Pair of Straight Lines
4.6. Problems
5. Circles
5.1. Special Cases
5.2. Circle on a Diameter
5.3. Parametric Form of a Circle
5.4. Position of a Point w.r.t. a Circle
5.5. Intersection of a Line and a Circle
5.6. Tangents and Normals
5.7. Condition that a Line Touches a Circle
5.7.1. Two Tangents to a Circle
5.7.2. Length of a Tangent
5.7.3. Pair of Tangents
5.8. Chord of Contact of Tangents
5.9. Poles and Polars
5.9.1. Coordinates of a Pole
5.9.2. Properties of Poles and Polars
5.10. Equation of a Chord
5.11. Intersection of Circles
5.11.1. Orthogonal Circles
5.12. Radical Axis
5.12.1. Properties of Radical Axes
5.13. Coaxial Circles
5.13.1. Limiting Points of a Coaxial System
5.13.2. Circles through the Points of Intersection of a Circle and a Line
5.14. Problems
6. Conic Sections
6.1. Definition
6.1.1. Equation of a Conic
6.2. Parabola
6.2.1. Intersection of a Line and a Parabola
6.2.2. Equation of a Tangent at a Given Point on a Parabola
6.2.3. Equation of a Normal at a Given Point on a Parabola
6.2.4. Equation of the Normal in terms of its Slope
6.2.5. Parametric Equation of the Parabola y 2 = 4 a x
6.2.6. Equation of a Chord of a Parabola
6.2.7. Tangent and Normal at any Point t
6.2.8. Equation of the Chord of a Parabola whose middle point is ( x 1 , y 1 )
6.2.9. Position of a Point w.r.t. a Parabola
6.2.10. Tangents from a Point outside a Parabola to the Parabola
6.2.11. Equation of Pair of Tangents from a Point to a Parabola
6.2.12. Equation of Chord of Contact of point P ( α , β ) w.r.t. a Parabola
6.2.13. Pole and Polar of a Parabola
6.2.14. Diameter of a Parabola
6.2.15. The Optical Property of a Parabola
6.3. Ellipse
6.3.1. Equation of an Ellipse
II.
7. Answers of Coordinates
8. Answers of locus
9. Answers of Straight Lines
10. Answers of Pair of Straight Lines
11. Answers of Circles
Index
A. GNU Free Documentation License
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  Preface

© 2026-present Shiv S. Dayal. ashtavakra.org. GNU FDL license v1.3 or later is applicable where not stated.