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Analytical Geometry

Book Description

Designed to meet the requirements of undergraduate students, Analytical Geometry: 2D and 3D deals with the theoretical as well as the practical aspects of the subject. Equal emphasis has been given to both 2D as well as 3D geometry. The book follows a systematic approach with adequate examples for better understanding of the concepts.

Table of Contents

  1. Cover
  2. Title Page
  3. Brief Content
  4. Contents
  5. About the Author
  6. Dedication
  7. Preface
  8. Chapter 1: Coordinate Geometry
    1. 1.1 Introduction
    2. 1.2 Section Formula
    3. Illustrative Examples
    4. Exercises
  9. Chapter 2: The Straight Line
    1. 2.1 Introduction
    2. 2.2 Slope of a Straight Line
    3. 2.3 Slope-intercept Form of a Straight Line
    4. 2.4 Intercept Form
    5. 2.5 Slope-point Form
    6. 2.6 Two Points Form
    7. 2.7 Normal Form
    8. 2.8 Parametric Form and Distance Form
    9. 2.9 Perpendicular Distance on a Straight Line
    10. 2.10 Intersection of Two Straight Lines
    11. 2.11 Concurrent Straight Lines
    12. 2.12 Angle between Two Straight Lines
    13. 2.13 Equations of Bisectors of the Angle between Two Lines
    14. Illustrative Examples
    15. Exercises
  10. Chapter 3: Pair of Straight Lines
    1. 3.1 Introduction
    2. 3.2 Homogeneous Equation of Second Degree in x and y
    3. 3.3 Angle between the Lines Represented by ax2 + 2hxy + by2 = 0
    4. 3.4 Equation for the Bisector of the Angles between the Lines Given by ax2 + 2hxy + by2 = 0
    5. 3.5 Condition for General Equation of a Second Degree Equation to Represent a Pair of Straight Lines
    6. Illustrative Examples
    7. Exercises
  11. Chapter 4: Circle
    1. 4.1 Introduction
    2. 4.2 Equation of a Circle whose Centre is (h, k) and Radius r
    3. 4.3 Centre and Radius of a Circle Represented by the Equation x2 + y2 + 2gx + 2fy + c = 0
    4. 4.4 Length of Tangent from Point P(x1, y1) to the Circle x2 + y2 + 2gx + 2fy + c = 0
    5. 4.5 Equation of Tangent at (x1, y1) to the Circle x2 + y2 + 2gx + 2fy + c = 0
    6. 4.6 Equation of Circle with the Line Joining Points A (x1, y1) and B (x2, y2) as the ends of Diameter
    7. 4.7 Condition for the Straight Line y = mx + c to be a Tangent to the Circle x2 + y2 = a2
    8. 4.8 Equation of the Chord of Contact of Tangents from (x1, y1) to the Circle x2 + y2 + 2gx + 2fy + c = 0
    9. 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
    10. 4.10 Pole and Polar
    11. 4.11 Conjugate Lines
    12. 4.12 Equation of a Chord of Circle x2 + y2 + 2gx + 2fy + c = 0 in Terms of its Middle Point
    13. 4.13 Combined Equation of a Pair of Tangents from (x1, y1) to the Circle x2 + y2 + 2gx + 2fy + c = 0
    14. 4.14 Parametric Form of a Circle
    15. Illustrative Examples
    16. Exercises
  12. Chapter 5: System of Circles
    1. 5.1 Radical Axis of Two Circles
    2. 5.2 Orthogonal Circles
    3. 5.3 Coaxal System
    4. 5.4 Limiting Points
    5. 5.5 Examples (Radical Axis)
    6. 5.6 Examples (Limiting Points)
    7. Exercises
  13. Chapter 6: Parabola
    1. 6.1 Introduction
    2. 6.2 General Equation of a Conic
    3. 6.3 Equation of a Parabola
    4. 6.4 Length of Latus Rectum
    5. 6.5 Different Forms of Parabola
    6. Illustrative Examples Based on Focus Directrix Property
    7. 6.6 Condition for Tangency
    8. 6.7 Number of Tangents
    9. 6.8 Perpendicular Tangents
    10. 6.9 Equation of Tangent
    11. 6.10 Equation of Normal
    12. 6.11 Equation of Chord of Contact
    13. 6.12 Polar of a Point
    14. 6.13 Conjugate Lines
    15. 6.14 Pair of Tangents
    16. 6.15 Chord Interms of Mid-point
    17. 6.16 Parametric Representation
    18. 6.17 Chord Joining Two Points
    19. 6.18 Equations of Tangent and Normal
    20. 6.19 Point of Intersection of Tangents
    21. 6.20 Point of Intersection of Normals
    22. 6.21 Number of Normals from a Point
    23. 6.22 Intersection of a Parabola and a Circle
    24. Illustrative Examples Based on Tangents and Normals
    25. Illustrative Examples Based on Parameters
    26. Exercises
  14. Chapter 7: Ellipse
    1. 7.1 Standard Equation
    2. 7.2 Standard Equation of an Ellipse
    3. 7.3 Focal Distance
    4. 7.4 Position of a Point
    5. 7.5 Auxiliary Circle
    6. Illustrative Examples Based on Focus-directrix Property
    7. 7.6 Condition for Tangency
    8. 7.7 Director Circle of an Ellipse
    9. 7.8 Equation of the Tangent
    10. 7.9 Equation of Tangent and Normal
    11. 7.10 Equation to the Chord of Contact
    12. 7.11 Equation of the Polar
    13. 7.12 Condition for Conjugate Lines
    14. Illustrative Examples Based on Tangents, Normals, Pole-polar and Chord
    15. 7.13 Eccentric Angle
    16. 7.14 Equation of the Chord Joining the Points
    17. 7.15 Equation of Tangent at ‘θ’ on the Ellipse
    18. 7.16 Conormal Points
    19. 7.17 Concyclic Points
    20. 7.18 Equation of a Chord in Terms of its Middle Point
    21. 7.19 Combined Equation of Pair of Tangents
    22. 7.20 Conjugate Diameters
    23. 7.21 Equi-conjugate Diameters
    24. Illustrative Examples Based on Conjugate Diameters
    25. Exercises
  15. Chapter 8: Hyperbola
    1. 8.1 Definition
    2. 8.2 Standard Equation
    3. 8.3 Important Property of Hyperbola
    4. 8.4 Equation of Hyperbola in Parametric Form
    5. 8.5 Rectangular Hyperbola
    6. 8.6 Conjugate Hyperbola
    7. 8.7 Asymptotes
    8. 8.8 Conjugate Diameters
    9. 8.9 Rectangular Hyperbola
    10. Exercises
  16. Chapter 9: Polar Coordinates
    1. 9.1 Introduction
    2. 9.2 Definition of Polar Coordinates
    3. 9.3 Relation between Cartesian Coordinates and Polar Coordinates
    4. 9.4 Polar Equation of a Straight Line
    5. 9.5 Polar Equation of a Straight Line in Normal Form
    6. 9.6 Circle
    7. 9.7 Polar Equation of a Conic
    8. Exercises
  17. Chapter 10: Tracing of Curves
    1. 10.1 General Equation of the Second Degree and Tracing of a Conic
    2. 10.2 Shift of Origin without Changing the Direction of Axes
    3. 10.3 Rotation of Axes without Changing the Origin
    4. 10.4 Removal of XY-term
    5. 10.5 Invariants
    6. 10.6 Conditions for the General Equation of the Second Degree to Represent a Conic
    7. 10.7 Centre of the Conic Given by the General Equation of the Second Degree
    8. 10.8 Equation of the Conic Referred to the Centre as Origin
    9. 10.9 Length and Position of the Axes of the Central Conic whose Equation is ax2 + 2hxy + by2 = 1
    10. 10.10 Axis and Vertex of the Parabola whose Equation is ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
    11. Exercises
  18. Chapter 11: Three Dimension
    1. 11.1 Rectangular Coordinate Axes
    2. 11.2 Formula for Distance between Two Points
    3. 11.3 Centroid of Triangle
    4. 11.4 Centroid of Tetrahedron
    5. 11.5 Direction Cosines
    6. Illustrative Examples
    7. Exercises
  19. Chapter 12: Plane
    1. 12.1 Introduction
    2. 12.2 General Equation of a Plane
    3. 12.3 General Equation of a Plane Passing Through a Given Point
    4. 12.4 Equation of a Plane in Intercept Form
    5. 12.5 Equation of a Plane in Normal Form
    6. 12.6 Angle between Two Planes
    7. 12.7 Perpendicular Distance from a Point on a Plane
    8. 12.8 Plane Passing Through Three Given Points
    9. 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.
    10. 12.10 Plane Passing Through the Intersection of Two Given Planes
    11. 12.11 Equation of the Planes which Bisect the Angle between Two Given Planes
    12. 12.12 Condition for the Homogenous Equation of the Second Degree to Represent a Pair of Planes
    13. Illustrative Examples
    14. Exercises
  20. Chapter 13: Straight Line
    1. 13.1 Introduction
    2. 13.2 Equation of a Straight Line in Symmetrical Form
    3. 13.3 Equations of a Straight Line Passing Through the Two Given Points
    4. 13.4 Equations of a Straight Line Determined by a Pair of Planes in Symmetrical Form
    5. 13.5 Angle between a Plane and a Line
    6. 13.6 Condition for a Line to be Parallel to a Plane
    7. 13.7 Conditions for a Line to Lie on a Plane
    8. 13.8 To Find the Length of the Perpendicular from a Given Point on a Line
    9. 13.9 Coplanar Lines
    10. 13.10 Skew Lines
    11. 13.11 Equations of Two Non-intersecting Lines
    12. 13.12 Intersection of Three Planes
    13. 13.13 Conditions for Three Given Planes to Form a Triangular Prism
    14. Illustrative Examples
    15. Illustrative Examples (Coplanar Lines and Shortest Distance)
    16. Exercises
  21. Chapter 14: Sphere
    1. 14.1 Definition of Sphere
    2. 14.2 The equation of a sphere with centre at (a, b, c) and radius r
    3. 14.3 Equation of the Sphere on the Line Joining the Points (x1, y1, z1) and (x2, y2, z2) as Diameter
    4. 14.4 Length of the Tangent from P(x1, y1, z1) to the Sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0
    5. 14.5 Equation of the Tangent Plane at (x1, y1, z1) to the Sphere x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0
    6. 14.6 Section of a Sphere by a Plane
    7. 14.7 Equation of a Circle
    8. 14.8 Intersection of Two Spheres
    9. 14.9 Equation of a Sphere Passing Through a Given Circle
    10. 14.10 Condition for Orthogonality of Two Spheres
    11. 14.11 Radical Plane
    12. 14.12 Coaxal System
    13. Illustrative Examples
    14. Exercises
  22. Chapter 15: Cone
    1. 15.1 Definition of Cone
    2. 15.2 Equation of a Cone with a Given Vertex and a Given Guiding Curve
    3. 15.3 Equation of a Cone with its Vertex at the Origin
    4. 15.4 Condition for the General Equation of the Second Degree to Represent a Cone
    5. 15.5 Right Circular Cone
    6. 15.6 Tangent Plane
    7. 15.7 Reciprocal Cone
    8. Exercises
  23. Chapter 16: Cylinder
    1. 16.1 Definition
    2. 16.2 Equation of a Cylinder with a Given Generator and a Given Guiding Curve
    3. 16.3 Enveloping Cylinder
    4. 16.4 Right Circular Cylinder
    5. Illustrative Examples
    6. Exercises
  24. Acknowledgement
  25. Copyright
  26. Back Cover