An Elementary Course in Analytic Geometry

Front Cover
American Book Company, 1898 - Geometry, Analytic - 418 pages
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Contents

2 Cartesian co÷rdinates axes not rectangular
32
3 Rectangular co÷rdinates 27 Slope of a line
33
Summary 29 The area of a triangle 1 Rectangular co÷rdinates
34
2 Polar co÷rdinates
36
To find the co÷rdinates of the point which divides in a given ratio the straight line from one given point to another
37
Fundamental problems of analytic geometry
40
33
43
Equation of straight line through given point and in given
44
Loci by polar co÷rdinates
46
The locus of an equation 36 Classification of loci 37 Construction of loci Discussion of equations
49
a by any trans position of the terms of the equation and B by multiply ing both members of the equation by any finite constant
52
Points of intersection of two loci
53
Product of two or more equations
54
Locus represented by the sum of two equations
56
43
61
46
65
LESSONS
71
ABTICLE
74
CHAPTER V
81
Equation of a normal to a given circle
85
second method
87
To trace the locus of an equation of the first degree
94
Line which makes a given angle with a given line
101
The distance of a given point from a given line
107
The ellipse defined
109
co÷rdinate axes oblique
115
Equation of a chord of contact
126
31
134
CHAPTER VII
135
Illustrative examples
141
Lengths of tangents and normals Subtangents and sub normals 87 Tangent and normal lengths subtangent and subnormal for the circle
149
To find the length of a tangent from a given external point to a given circle
151
From any point outside of a circle two tangents to the circle can be drawn
152
Chord of contact
155
Poles and polars 92 Equation of the polar
156
Fundamental theorem
157
Geometrical construction for the polar of a given point and for the pole of a given line with regard to a given circle
158
Circles throușh the intersections of two given circles 96 Common chord of two circles
160
Radical axis radical center
161
polar co÷rdinates
162
Equation of a circle referred to oblique axes
163
The angle formed by two intersecting curves 152 154 156 156
164
CHAPTER VIII
170
First standard form of the equation of the parabola
171
To trace the parabola ya 4 px 105 Latus rectum 106 Geometric property of the parabola Second standard equa
173
Every equation of the form Ax2 + 2 Gx + 2 Fy + C 0 or By2 + 2 Gx + 2 Fy + C
175
2
177
Intrinsic property of the ellipse Second standard equation
183
Reduction of the equation of an ellipse to a standard form
189
Definition
193
The spiral of Archimedes
194
Intrinsic property of the hyperbola Second standard equa
195
The parabolic spiral
196
The lituus or trumpet
197
The logarithmic spiral
198
Normal to the conic Ax▓ + By▓ + 2 Gx + 2 Fy + C 0 at
203
32
238
144
239
Construction of the ellipse
240
The tangent and normal bisect externally and internally
246
Diameters
253
Supplemental chords
259
CHAPTER XI
265
Conjugate hyperbolas
271
Equilateral or rectangular hyperbola
277
172
285
173
287
Equations representing an hyperbola but involving only one variable LAGE 284 285
288
175
292
Illustrative examples
294
177
297
Center of a conic section
298
183 The equation of a conic through given points
299
5
300
HIGHER PLANE CURVES 184 Definitions
309
The conchoid of Nicomedes
312
The witch of Agnesi
314
The lemniscate of Bernouilli
315
a The limašon of Pascal
318
b The cardioid
319
The Neilian or semicubical parabola 309 312 314 315 318 319 320
320
Transcendental Curves
321
The cycloid
323
The hypocycloid
325
SOLID ANALYTIC GEOMETRY CHAPTER I
331
333
333
direction cosines
334
Distance and direction from one point to another rectangu lar co÷rdinates
336
The point which divides in a given ratio the straight line from one point to another
337
Angle between two radii vectores Angle between two lines 207 Transformation of co÷rdinates rectangular systems
338
THE Locus OF AN EQUATION SURFACES
342
Introductory 209 Equations in one variable Planes parallel to co÷rdinate planes
343
Equations in two variables Cylinders perpendicular to co÷r dinate planes
344
Equations in three variables Surfaces
346
Curves Traces of surfaces
347
Surfaces of revolution
348
EQUATIONS OF THE FIRST DEGREE Ax By + Cz + D 0 PLANES
353
Distance of a point from a plane
359
CHAPTER IV
367
33
370
12
380
NOTE A Historical sketch
381
NOTE E Parabola as a limiting form of ellipse or hyperbola
387
82
391
61
393
321
402
262264
409
40
415
Co÷rdinates of a point
416
323
417
348
418
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Page 120 - The straight line joining the middle points of two sides of a triangle is parallel to the third side, and equal to half of it.
Page 236 - Find the locus of the center of a circle which passes through a given point and touches a given line.
Page 108 - 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 170 - Thus a parabola is the locus of a point which moves so that its distance from a fixed point is equal to its distance from a fixed straight line (see fig.
Page 179 - F') ; the diameter drawn through them is called the major axis, and the perpendicular bisector of this diameter the minor axis. It is also defined as the locus of a point which moves so that the ratio of its distance from a fixed point...
Page 67 - A conic section or conic is 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 straight line...
Page 240 - Art. 144 is sometimes given as the definition of the ellipse ; viz. the ellipse is the locus of a point the sum of whose distances from two fixed points is constant.
Page 122 - The line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of the third side.
Page 211 - To draw that diameter of a given circle which shall pass at a given distance from a given point. 9. Find the locus of the middle points of any system of parallel chords in a circle.
Page 169 - A point moves so that the square of its distance from the base of an isosceles triangle is equal to the product of its distances from the other two sides.

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