Analytical Geometry, 1/e: 2D and 3D

Cover......Page 1
Dedication......Page 6
Brief Contents......Page 8
Contents......Page 10
About the Author......Page 20
Preface......Page 22
 1.1 Introduction......Page 24
  1.1.1 Distance between Two Given Points......Page 25
  1.2.1 Coordinates of the Point that Divides the Line Joining Two Given Points in a Given Ratio......Page 26
  1.2.3 Centroid of a Triangle Given its Vertices......Page 27
  1.2.4 A rea of Triangle ABC with Vertices A(x1, y1), B(x2, y2) and C(x3, y3)......Page 28
  1.2.5 Area of the Quadrilateral Given its Vertices......Page 29
 Illustrative Examples......Page 30
 Exercises......Page 50
  2.1.1 Determination of the General Equation of a Straight Line......Page 54
  2.1.2 Equation of a Straight Line Parallel to y-axis and at a Distance of h units from x-axis......Page 55
 2.3 Slope-intercept Form of a Straight Line......Page 56
 2.4 Intercept Form......Page 57
 2.7 Normal Form......Page 58
 2.8 Parametric Form and Distance Form......Page 59
 2.9 Perpendicular Distance on a Straight Line......Page 60
 2.10 Intersection of Two Straight Lines......Page 61
 2.11 Concurrent Straight Lines......Page 63
 2.12 Angle between Two Straight Lines......Page 64
 Illustrative Examples......Page 65
 Exercises......Page 98
 3.1 Introduction......Page 106
 3.3 Angle between the Lines Represented by ax 2 + 2hxy + by 2 = 0......Page 107
 3.4 Equation for the Bisector of the Angles between the Lines Given by ax 2 + 2hxy + by 2 = 0......Page 108
 3.5 Condition for General Equation of a Second Degree Equation to Represent a Pair of Straight Lines......Page 110
 Illustrative Examples......Page 114
 Exercises......Page 140
 4.3 Centre and Radius of a Circle Represented by the Equation x 2 + y 2 + 2gx + 2fy + c = 0......Page 142
 4.4 Length of Tangent from Point P(x1, y1) to the Circle x 2 + y 2 + 2gx + 2fy + c = 0......Page 143
 4.6 Equation of Circle with the Line Joining Points A (x1, y1) and B (x2, y2) as the ends of Diameter......Page 144
 4.7 Condition for the Straight Line y = mx + c to be a Tangent to the Circle x 2 + y 2 = a 2......Page 145
 4.8 Equation of the Chord of Contact of Tangents from (x1, y1) to the Circle x 2 + y 2 + 2gx + 2fy + c = 0......Page 146
 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......Page 147
  4.10.1 Polar of the Point P (x1, y1) with Respect to the Circle x 2 + y 2 + 2gx + 2fy + c = 0......Page 148
  4.11.1 Condition for the Lines lx + my + n = 0 and l1x + m1y + n1 = 0 to be Conjugate Lines with Respect to the Circle x 2 + y 2 = a2......Page 149
 4.12 Equation of a Chord of Circle x 2 + y 2 + 2gx + 2fy + c = 0 in Terms of its Middle Point......Page 150
 4.13 Combined Equation of a Pair of Tangents from (x1, y1) to the Circle x 2 + y 2 + 2gx + 2fy + c = 0......Page 151
 Illustrative Examples......Page 152
 Exercises......Page 187
 5.1 Radical Axis of Two Circles......Page 198
 5.2 Orthogonal Circles......Page 199
 5.3 Coaxal System......Page 200
 5.4 Limiting Points......Page 201
 5.5 Examples (Radical Axis)......Page 203
 5.6 Examples (Limiting Points)......Page 215
 Exercises......Page 223
 6.2 General Equation of a Conic......Page 226
 6.3 Equation of a Parabola......Page 227
 6.5 Different Forms of Parabola......Page 228
 Illustrative Examples Based on Focus Directrix Property......Page 230
 6.6 Condition for Tangency......Page 233
 6.8 Perpendicular Tangents......Page 234
 6.9 Equation of Tangent......Page 235
 6.10 Equation of Normal......Page 236
 6.11 Equation of Chord of Contact......Page 237
 6.13 Conjugate Lines......Page 238
 6.14 Pair of Tangents......Page 239
 6.15 Chord Interms of Mid-point......Page 240
 6.17 Chord Joining Two Points......Page 241
 6.19 Point of Intersection of Tangents......Page 242
 6.21 Number of Normals from a Point......Page 243
 6.22 Intersection of a Parabola and a Circle......Page 244
 Illustrative Examples Based on Tangents and Normals......Page 245
 Illustrative Examples Based on Parameters......Page 266
 Exercises......Page 287
 7.2 Standard Equation of an Ellipse......Page 290
 7.4 Position of a Point......Page 293
 7.5 Auxiliary Circle......Page 294
 Illustrative Examples Based on Focus-directrix Property......Page 295
 7.6 Condition for Tangency......Page 300
 7.7 Director Circle of an Ellipse......Page 301
 7.8 Equation of the Tangent......Page 302
 7.9 Equation of Tangent and Normal......Page 303
 7.10 Equation to the Chord of Contact......Page 305
 7.11 Equation of the Polar......Page 306
 7.12 Condition for Conjugate Lines......Page 307
 Illustrative Examples Based on Tangents, Normals, Pole-polar and Chord......Page 308
 7.14 Equation of the Chord Joining the Points......Page 334
 7.15 Equation of Tangent at ‘p ’ on the Ellipse......Page 336
 7.16 Conormal Points......Page 337
 7.17 Concyclic Points......Page 338
 7.18 Equation of a Chord in Terms of its Middle Point......Page 340
 7.19 Combined Equation of Pair of Tangents......Page 341
 7.20 Conjugate Diameters......Page 343
  7.20.1 Locus of Midpoint......Page 369
  7.20.2 Property: The Eccentric Angles of the Extremities of a Pair of Semi-conjugate Diameter Differ by a Right Angle......Page 370
  7.20.4 Property: The Tangents at the Extremities of a Pair of Conjugate Diameters of an Ellipse Encloses a Parallelogram Whose Area Is Constant......Page 371
  7.20.5 Property: The Product of the Focal Distances of a Point on an Ellipse Is Equal to the Square of the Semi-diameter Which Is Conjugate to the Diameter Through the Point......Page 372
 7.21.1 Property: Equi-conjugate Diameters of an Ellipse Lie along the Diagonals of the Rectangle Formed by the Tangent at the Ends of its Axes......Page 373
 Illustrative Examples Based on Conjugate Diameters......Page 374
 Exercises......Page 389
 8.2 Standard Equation......Page 394
 8.5 Rectangular Hyperbola......Page 396
 8.6 Conjugate Hyperbola......Page 397
  8.7.1 Equations of Asymptotes of the Hyperbola......Page 421
  8.7.2 Angle between the Asymptotes......Page 423
 8.8 Conjugate Diameters......Page 428
  8.9.1 Equation of Rectangular Hyperbola with Reference to Asymptotes as Axes......Page 434
  8.9.2 Equations of Tangent and Normal at (x1, y1) on the Rectangular Hyperbola xy = c 2......Page 435
  8.9.3 Equation of Tangent and Normal at ctct, °  on the Rectangular Hyperbola xy = c 2......Page 436
  8.9.6 Results Concerning the Rectangular Hyperbola......Page 437
  8.9.8 Concyclic Points on the Rectangular Hyperbola......Page 438
 Exercises......Page 454
 9.2 Definition of Polar Coordinates......Page 458
 9.3 Relation between Cartesian Coordinates and Polar Coordinates......Page 460
 9.5 Polar Equation of a Straight Line in Normal Form......Page 461
  9.6.1 Polar Equation of a Circle......Page 466
  9.6.2 Equation of the Chord of the Circle r = 2a cos p on the Line Joining the Points (r1, p1) and (r2, p2).......Page 468
  9.6.3 Equation of the Normal at ` on the Circle r = 2` cosp......Page 469
  9.6.4 Equation of the Circle on the Line Joining the Points (a, ` ) and (b, a ) as the ends of a Diameter......Page 470
  9.7.1 Polar Equation of a Conic......Page 474
  9.7.2 Equation to the Directrix Corresponding to the Pole......Page 477
  9.7.3 Equation to the Directrix Corresponding to Focus Other than the Pole......Page 478
  9.7.4 Equation of Chord Joining the Points whose Vectorial Angles are ` - a and ` + a on the Conic......Page 479
  9.7.5 Tangent at the Point whose Vectorial Angle is ` on the Coniclr=1+ ecosp......Page 480
  9.7.6 Equation of Normal at the Point whose Vectorial Angle is a on the Conic......Page 481
  9.7.7 Asymptotes of the Conic islr=1+ecosp (e>1)......Page 482
  9.7.8 Equation of Chord of Contact of Tangents from (r1,p1) to the Conic......Page 483
  9.7.9 Equation of the Polar of any Point (r1,p1) with Respect to the Coniclr=1+ ecosp......Page 484
 Exercises......Page 508
 10.2 Shift of Origin without Changing the Direction of Axes......Page 510
 10.3 Rotation of Axes without Changing the Origin......Page 511
 10.5 Invariants......Page 512
 10.6 Conditions for the General Equation of the Second Degree to Represent a Conic......Page 513
 10.8 Equation of the Conic Referred to the Centre as Origin......Page 515
 10.9 Length and Position of the Axes of the Central Conic whose Equation is ax 2 + 2hxy + by 2 = 1......Page 517
 10.10 Axis and Vertex of the Parabola whose Equation is ax 2 + 2hxy + by 2 + 2gx + 2fy + c = 0......Page 519
 Exercises......Page 525
 11.1 Rectangular Coordinate Axes......Page 526
 11.2 Formula for Distance between Two Points......Page 529
  11.2.1 Section Formula......Page 530
 11.3 Centroid of Triangle......Page 532
 11.5 Direction Cosines......Page 533
  11.5.1 Direction Ratios......Page 534
  11.5.2 Projection of a Line......Page 535
  11.5.4 Angle between Two Given Lines......Page 536
 Illustrative Examples......Page 538
 Exercises......Page 550
 12.2 General Equation of a Plane......Page 554
 12.4 Equation of a Plane in Intercept Form......Page 555
 12.5 Equation of a Plane in Normal Form......Page 556
 12.7 Perpendicular Distance From a Point on a Plane......Page 558
 12.8 Plane Passing Through Three Given Points......Page 560
 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.......Page 561
 12.10 Plane Passing Through the Intersection of Two Given Planes......Page 562
 12.11 Equation of the Planes which Bisect the Angle between Two Given Planes......Page 563
 12.12 Condition for the Homogenous Equation of the Second Degree to Represent a Pair of Planes......Page 564
 Illustrative Examples......Page 565
 Exercises......Page 586
 13.2 Equation of a Straight Line in Symmetrical Form......Page 592
 13.3 Equations of a Straight Line Passing Through the Two Given Points......Page 594
 13.4 Equations of a Straight Line Determined by a Pair of Planes in Symmetrical Form......Page 595
 13.5 Angle between a Plane and a Line......Page 596
 13.7 Conditions for a Line to Lie on a Plane......Page 597
 13.9 Coplanar Lines......Page 598
  13.10.1 Length and Equations of the Line of the Shortest Distance......Page 600
  13.10.2 Equation of the Line of SD......Page 603
 13.11 Equations of Two Non-intersecting Lines......Page 604
 13.12 Intersection of Three Planes......Page 605
 13.13 Conditions for Three Given Planes to Form a Triangular Prism......Page 607
 Illustrative Examples......Page 608
 Illustrative Examples (Coplanar Lines and Shortest Distance)......Page 625
 Exercises......Page 638
 14.2 The Equation of a Sphere with Centre at (a, b, c ) and Radius r......Page 644
 14.3 Equation of the Sphere on the Line Joining the Points (x 1, y 1, z 1) and (x 2, y 2, z 2) as Diameter......Page 645
 14.5 Equation of the Tangent Plane at (x1, y1, z1) to the Sphere x 2 + y 2 + z 2 + 2ux + 2vy + 2wz + d = 0......Page 646
 14.6 Section of a Sphere by a Plane......Page 647
 14.8 Intersection of Two Spheres......Page 648
 14.10 Condition for Orthogonality of Two Spheres......Page 649
  14.11.1 O btain the Equations to the Radical Plane of Two Given Spheres......Page 650
  14.11.2 Properties of Radical Plane......Page 651
  14.12.1 General Equation to a System of Coaxal Spheres......Page 652
  14.12.3 Limiting Points......Page 653
 Illustrative Examples......Page 654
 Exercises......Page 690
 15.2 Equation of a Cone with a Given Vertex and a Given Guiding Curve......Page 696
 15.3 Equation of a Cone with its Vertex at the Origin......Page 699
 15.4 Condition for the General Equation of the Second Degree to Represent a Cone......Page 702
 15.5 Right Circular Cone......Page 708
  15.5.1 Equation of a R ight Circular Cone with V ertex V (`, a, f ), Axis VL with Direction Ratios l, m, n and Semi-vertical Angle p......Page 709
  15.5.2 Enveloping Cone......Page 710
 15.6 Tangent plane......Page 718
  15.6.1 Condition for the Tangency of a Plane and a Cone......Page 720
  15.7.1 Equation of the Reciprocal Cone......Page 721
  15.7.2 Angle between Two Generating Lines in Which a Plane Cuts a Cone......Page 722
  15.7.3 Condition for Mutually Perpendicular Generators of the Cone......Page 725
 Exercises......Page 729
 16.2 Equation of a Cylinder with a Given Generator and a Given Guiding Curve......Page 730
 16.3 Enveloping Cylinder......Page 731
 16.4 Right Circular Cylinder......Page 732
 Illustrative Examples......Page 733
 Exercises......Page 747
Index......Page 750