Cartesian Plane Geometry, Volume 1

Front Cover
J.M. Dent & Company, 1907 - Conic sections - 428 pages
 

Selected pages

Contents

Harmonic division
9
Parallel lines Examples
10
Coordinates of a point
11
Distance between two points Examples
12
Coordinates of a line
13
Distance from the origin to a line
14
Distance from a point to a line Examples PAGE 3 1 1 2 3 4 5 8 10
15
Algebraic distinction between two sides of a line
16
Condition of incidence of a point and a line
17
Coordinates of line through two points of point on two lines
18
Point at infinity Line through origin
19
Collinear points and concurrent lines
20
Point that divides a segment in a given ratio
21
Proof without use of linecoordinates that a line is represented
26
Examples 121
32
CHAPTER IIIREPRESENTATION OF POINT AND LINE BY EQUATIONS
36
Equation of a line through a given point Examples
38
An equation of a line or point is of the first degree 24 An equation of the first degree represents a line or point 25 Relation between equation and coo...
51
Dualistic form
58
17
64
Statement of process for finding equation of envelope of
81
22
86
12828 29 31 33 25 25 26
87
36
90
SECTION PAGE 60 Common points of two loci
97
Interpretation of u + kv 0
98
Common lines of two envelopes
99
Lines that join the origin to the common points of two loci Examples
100
Imaginary points
104
CHAPTER VCONICS 68 Definition of a conic Equation is of the second degree Examples
108
Forms of equation of a conic
109
Ellipse and hyperbola Definition and simplest equation
110
Axes of symmetry Vertices
114
Lengths of axes Latus rectum central conics
115
Circle as a particular case of the ellipse
118
Examples 15
120
Relation connecting lengths of conjugate diameters of
121
CHAPTER VIRELATION OF STRAIGHT LINES TO CURVES 76 Order of a locus class of an envelope Examples
126
Intersections of a line with a conic Examples
127
Definition of tangent and normal
129
Condition that y mx + n be a tangent to the parabola circle ellipse hyperbola Examples
131
Tangents with a given slope Asymptotes
135
Relation of a hyperbola to its asymptotes
137
Examples on tangents Examples
138
Slopes of tangents from a point Examples
140
Tangents from a point are real or imaginary
143
Power of a point with respect to a curve
144
Tangents from a point are real or imaginary according to sign of power of the point Examples
148
Condition that a line be a tangent
150
Lineequation of parabola ellipse hyperbola
151
Use of lineequation of a conic in finding tangents from a point Examples
152
Equation of pair of tangents from a point
154
Pole of a line Examples
172
Conjugate points or lines
173
Harmonic properties of poles and polars
174
Collinear poles give concurrent polars
176
Condition that points or lines be conjugate
177
Examples 120
180
Reciprocal polars treated by means of linecoordinates
183
Reciprocal polars treated without linecoordinates
184
Examples 18
185
Examples 111
187
37
189
Locus of points of bisection of parallel chords of a conic is a straight line
190
Ex 10 Four normals pass through a point they
193
Diameters
195
Extremities of a diameter
197
Geometrical proofs of theorems on diameters
198
Point of bisection of chord through a fixed point Examples
200
Conjugate diameters of central conics
201
Conjugate diameters of an ellipse lie in different quadrants
202
Conjugate diameters of a hyperbola lie in the same quadrant
203
Coordinates of extremities of conjugate diameters of an ellipse
204
The sum of the squares of conjugate diameters of an ellipse is constant
207
Equiconjugate diameters
208
Coordinates of extremities of conjugate diameters of a hyperbola
209
The parabola has no asymptotes
218
The part of the tangent between the asymptotes is bisected
224
Conjugate hyperbolas
231
The parabola Formulæ and equations
238
The ellipse Formulæ and equations
249
The ellipse Properties and theorems
257
The hyperbola Formulæ and equations
270
The hyperbola Properties and theorems
276
The circle Formulæ and equations
282
SECTION PAGE
289
Object of changing axes
295
Proof that every equation of the second degree represents
305
Special process of reduction for the parabola
312
Determination of the focus and directrix of a parabola
318
CHAPTER XIISYSTEMS OF CONICS
330
The isotropic lines or null lines
337
Confocal conics
348
Locus of a point in a given relation to the conics of a system
354
Polar reciprocals
361
Ex 3 Three normals to a parabola pass through a point
367
Ex 8 Orthocentre centroid and circumcentre of a conormal
374
Ex 13 Slopes of the four normals from a given point
382
Ex 18 Condition satisfied by the parameters of three
392
Ex 20 Conics through four points Relation satisfied by
398
Ex 22 Equilateral triangle inscribed to a central conic
404
The graphical solution of equations
412
Ex 26 Determination of a rectangular hyperbola that cuts
419
Equations of circles that cut a given rectangular hyperbola
426

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Page 109 - A conic section is the locus of a point which moves so that its distance from a fixed point, called the focus, is in a constant ratio to its distance from a fixed straight line, called the directrix.
Page 236 - The parabola is the locus of a point whose distance from a fixed point, the focus, is always equal to its distance from a fixed line, the directrix.
Page 121 - PF'/PH'= e, by the definition of the curve. Furthermore :J (b) \PF—PF'\=2a. In fact, the hyperbola is often defined as the locus of a point which moves so that the difference of its distances from two fixed points is constant.
Page 121 - Find the locus of a point which moves so that the sum of its distances from two vertices of an equilateral triangle shall equal its distance from the third.
Page 95 - PF'/PH' = e, by definition of the curve. Furthermore :f (6) PF + PF' = 2a. In fact, the ellipse is often defined as the locus of a point which moves so that the sum of its distances from two fixed points is constant.
Page 33 - A point moves so that the sum of the squares of its distances from the points (0, 0), (1, 0) is constant.
Page 109 - ... line. The fixed point is called the focus, the fixed line is called the directrix; the constant ratio is called the eccentricity, and is denoted by e.
Page 409 - The foot of the perpendicular from the focus of a parabola to a tangent lies on the tangent at the vertex.
Page 94 - What is the locus of a point which moves so that the square of its distance from a fixed point is proportional to its distance from a fixed line?