《ANALYTICAL CONICS VOLUME 3》

Ⅰ．INTRODUCTION1

1．Point coordinates1

2．Distance between two points2

3．Area of a triangle2

4．Joachimsthal＇s section formula4

5．Harmonic range of points5

Ⅱ．STRAIGHT LINE7

6．Straight line7

7．Parametric form8

8．Angle between two straight lines9

9．Intersection of two straight lines10

10．Concurrency11

11．Pencil of straight lines12

12．Sign of the expression u=lx+my+n13

13．Perpendicular distance of a point from a straight line14

14．Bisectors of angles between two straight lines15

15．Condition that ax2+2hxy+by2+2gx+2fy+c=0 represents two straight lines16

16．Angle between line-pair S=017

17．Equation of lines joining origin to the points of intersection of a line-pair and a straight line18

Ⅲ．CIRCLE21

18．Equation of circle21

19．Conjugate points22

20．Pole and polar24

21．Conjugate lines25

22．Tangent and normal25

23．Condition that a straight line touch a circle26

24．Pair of tangents from a point27

25．Angle of intersection of two circles28

26．Radical axis28

27．Coaxai circles30

28．Conjugate system of coaxal circles31

Ⅳ．ELLIPSE34

29．Ellipse34

30．Tangent and polar properties35

31．Orthoptic locus36

32．Parametric equations36

33．Conjugate diameters38

34．Focus and directrix40

35．Normal42

36．Concycllc points43

Ⅴ．HYPERBOLA45

37．Hyperbola45

38．Tangent and polar properties46

39．Parametric equations46

40．Asymptotes48

41．Conjugate diameters49

42．Focus and directrix50

43．Rectangular hyperbola51

Ⅵ．PARABOLA54

44．Parabola54

45．Tangent and polar properties55

46．Parametric equations55

47．Diameters56

48．Normals57

Ⅶ．TRANSFORMATION OF AXES58

49．Translation of axes58

50．Invariauts of a quadratic form under translation59

51．Rotation of axes60

52．Invariants of a quadratic form under rotation61

53．General transformation of axes62

54．Absolute invariants64

Ⅷ．GENERAL EQUATION OF THE SECOND DEGREE66

55．General quadratic equation66

56．Central conic68

57．Parabola69

58．Line-pair69

59．Classification of conic69

Ⅸ．LINE AT INFINITY71

60．Homogeneous cartesian coordinates71

61．Point at infinity71

62．Straight lines72

63．Circular points at infinity73

64．Conic and line at infinity74

65．Joachimsthal＇s section formula74

Ⅹ．CONIC76

66．Conic76

67．Cofactor notation76

68．Joachimsthal＇s quadratic equation77

69．Polar properties79

70．Tangent properties80

71．Orthoptic locus82

72．Centre83

73．Conjugate diameters83

74．Principal axes84

75．Asymptotes86

76．Focus and directrix86

77．Eccentricity89

Ⅺ．PENCIL OF CONICS91

78．Pencil of conics91

79．Contact of conics92

80．Circle of curvature94

81．Pair of tangents at ends of a chord95

Ⅻ．HOMOGRAPHIC CORRESPONDENCE97

82．Cross-ratio97

83．Harmonic case98

84．Homographic correspondence98

85．Fundamental cross-ratio theorem99

86．Involution100

87．Range of points101

88．Pencil of lines102

ⅩⅢ．LINE-COORDINATES105

89．Line-coordinates105

90．Line-conic106

91．Joachimsthal＇s quadratic equation107

92．Polar properties108

93．Intersections with conic109

94．Principle of duality110

95．Range of conics111

96．Foci111

97．Confocal conics113

98．Central confocal conics114

99．Confocal parabola115

ⅩⅣ．GENERALIZED HOMOGENEOUS COORDINATES118

100．Cofactor notation118

101．Generalized homogeneous coordinates118

102．Geometrical significance119

103．Range of points121

104．Straight line122

105．Pencil of lines122

106．Quadrangle123

107．Quadrilateral123

108．Arbitrary assignment of coordinates(±1,±1,±1)123

109．Conic124

110．Special forms of equation of conic125

111．Pencils of conics128

APPENDIX132

Worked-out solutions132

INDEX145