Analytic Geometry

Front Cover
Macmillan, 1927 - Geometry, Analytic - 257 pages
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Contents

The general equation of the first degree
41
Parallel and perpendicular lines
42
Equations containing arbitrary constants families of curves
44
Straight line determined by two conditions
45
The intercept form
47
The normal form
49
Distance of a point from a line
53
Angle between two lines
57
CHAPTER V
60
General equation
61
Point circles imaginary circles
62
Circle satisfying three conditions
64
Quadratic discriminant
67
Intersection of a line and a circle of two circles
68
Tangents to plane curves
70
Tangents having a given slope
72
Tangent at a given point of contact
74
Curves through the intersections of two given curves
76
Circle through the intersections of two given circles
78
Common chord
81
CHAPTER VI
84
Curve tracing in polar co÷rdinates
85
ART PAGE
87
SOLID ANALYTIC GEOMETRY
91
Definitions
94
Other standard forms
100
General equation
106
THE GENERAL EQUATION OF THE SECOND DEGREE
114
Conic through five points
121
ART PAGE 80 Parabola through four points
123
Polar equation of a conic
124
CHAPTER VIII
126
Geometric problems involving the parabola
128
Tangent at a given point
130
Normal subtangent subnormal
131
Tangent to the parabola y▓ 4ax
132
Subtangent and subnormal for the parabola
133
Tangents having a given slope tangents drawn from an external point
135
Two geometric properties
136
Chord of contact
138
Diameter of a conic
140
Diameter of a parabola
142
Geometric properties involving diameters
143
CHAPTER IX
147
A new definition of the ellipse
149
A new definition of the hyperbola
150
Tangent at a given point of contact
152
A property of the tangent to a central conic
154
Tangents having a given slope
155
Test for the species of a conic
170
Asymptotes
173
Single equation representing the pair of asymptotes
174
A general method for determination of asymptotes
178
Equilateral hyperbola referred to its asymptotes
181
CHAPTER X
182
Figures
183
Distance between two points
184
Direction angles direction cosines
187
Direction cosines proportional to three numbers
188
Direction cosines of the line through two points
189
Projections
191
Perpendicular lines
193
CHAPTER XI
194
Determination of points on a surface intercepts on the axes
195
The locus of two simultaneous equations
197
Intersection of three surfaces of a curve and a surface
198
CHAPTER XII
200
General form reduction to normal form
201
Perpendicular line and plane
202
Parallel planes
204
Plane through a given point
206
Angle between two planes
207
Planes perpendicular to a co÷rdinate plane
208
Intersection of three planes
210
CHAPTER XIII
211
Projecting planes
213
Symmetric form
215
Determination of direction cosines reduction to the sym metric form
216
Parallel intersecting and skew lines
217
Perpendicular line and plane
220
CHAPTER XIV
224
Sketching by parallel plane sections
225
Generation of a surface by a moving curve
226
The sphere
231
Quadric surfaces
233
The hyperboloid of one sheet
235
The hyperboloid of two sheets
236
The elliptic paraboloid
239
The hyperbolic paraboloid
240
Cylinders
242
Quadric cylinders
243
The elliptic cone
245
general case
246
CHAPTER XV
249
Plane sections of a quadric surface
253
INDEX
255
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Page 149 - Show that the locus of a point which moves so that the sum of its distances from two h'xed straight lines is constant is a straight line.
Page 91 - The locus of a point which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed line is called a conic.
Page 63 - A point moves so that the sum of the squares of its distances from two fixed points is constant.
Page 63 - A point moves so that its distances from two fixed points are in a constant ratio k.
Page 60 - A circle is the locus of a point at a constant distance from a fixed point. The fixed point is the center of the circle, and the constant distance is the radius.
Page 10 - N6 is to say that if two nonvertical lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other.
Page 92 - I, the conic is a parabola; if e > 1, the conic is a hyperbola. Every conic section is representable by an equation of second degree. Conversely, every equation of second degree in two variables represents a conic. Th...
Page 83 - The perpendicular bisectors of the sides of a triangle meet in a point. 12. The bisectors of the angles of a triangle meet in a point. 13. The tangents to a circle from an external point are equal. 14...
Page 31 - A point moves so that the sum of its distances from the two axes is always equal to 10.
Page 63 - Find the locus of a point which moves so that the square of its distance from the base of an isosceles triangle is equal to the rectangle under its distances from the other sides.

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