An Elementary Course in Analytic Geometry

Front Cover
American Book Company, 1898 - Geometry, Analytic - 390 pages
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Contents

The quadratic equation involving two unknowns
13
Trigonometric Conceptions and Formulas 13 Directed lines Angles
15
Trigonometric ratios
17
Functions of related angles
18
Other important formulas
19
ARTICLE PAGE
20
Orthogonal projection 11 12
21
15
23
CHAPTER II
24
Analytic Geometry
25
Cartesian co÷rdinates of points in a plane
26
Rectangular co÷rdinates
28
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
34
1 Rectangular coordinates 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 33
40
37
42
CHAPTER III
43
Equation of straight line through given point and in given
44
The locus of an equation
45
Cartesian co÷rdinates 34 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 equatio and B by multiply ing oth 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
Equation of a circle polar co÷rdinates
64
46
65
47
68
48
69
tions for each lesson The fifth day of each week should be devoted
70
49
73
Through a given external point two tangents to a conic
74
56
76
CHAPTER IV
79
Equation of straight line through two given points
81
CHAPTER V
81
Equation of a normal to a given circle
85
second method
87
Reduction of the general equation Ax + By + C 0 to
91
61
97
The distance of a given point from a given line
107
The ellipse defined
109
co÷rdinate axes oblique
115
PAGE
123
Equation of a chord of contact
126
The degree of an equation in Cartesian co÷rdinates is
129
CHAPTER VII
135
Illustrative examples
141
Construction of the ellipse
145
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 through 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
Definition
193
The spiral of Archimedes
194
The reciprocal or hyperbolic spiral
195
The parabolic spiral
196
Every equation of the form
197
The logarithmic spiral
198
CHAPTER IX
219
The subtangent and subnormal Construction of tangent
222
15
224
Some properties of the parabola which involve tangents
225
Some properties of the parabola involving diameters
232
18
236
The equation of the tangent to the ellipse
238
144
239
The tangent and normal bisect externally and internally
246
Diameters
253
Supplemental chords
259
CHAPTER XI
265
Asymptotes
275
Geometric properties of the hyperbola
281
171
284
172
285
173
287
Equations representing an hyperbola but involving only one variable
288
represents a parabola whose axis parallel to one of the co÷rdinate axes
290
Reduction of the equation of a parabola to a standard form 175
292
Illustrative examples
294
177
297
Center of a conic section
298
Transformation of the equation of a conic to parallel axes through its center
299
The invariants A + B and H2 AB
301
To reduce to its simplest standard form the general equation of a conic
303
The equation of a conic through given points a
307
CHAPTER XIII
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
190
320
Transcendental Curves 191 The cycloid
321
The hypocycloid
323
PAGE 325
325
195
326
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
337
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
353
Normal to the conic Ax▓ + By▓ + 2 Gx + 2 Fy + C 0 at
359
CHAPTER IV
367
equation
373
ANSWERS
379
Equation of a tangent and of a normal that pass through
405
344
416
348
418
and 69

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Page 104 - 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 220 - Find the locus of the center of a circle which passes through a given point and touches a given line.
Page 92 - 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 154 - 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 163 - 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 403 - The laboratory work is adapted to any equipment, and ihe instructions for it are placed in divisions by themselves, preceding the related chapters of descriptive text, which follows in the main the order of topics in Gray's Lessons in Botany. Special attention is paid to the ecological aspects of plant life, while at the same time morphology and physiology are fully treated. There are 384 carefully drawn illustrations, many of them entirely new.
Page 224 - 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 106 - 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 195 - 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 81 - A point moves so that the difference of the squares of its distances from two fixed points is constant. Show that the locus is a straight line. Hint. Draw XX' through the fixed points, and YY/ through their middle point.

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