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Page 46. The equation to a line, making a given angle with a given line, is

B
y - Y1 = (x − x1)

. 27 Itaß 47. If two lines y = ax + b and y =á x + b'are perpendicular to each other, we have lta = , or the lines are

1
y = 8 x *.b and y = 2 + 6

27 48. If p be the perpendicular from a given point (x1 yı) on the line y = x +b, then - 6

28 vi + 49. The length of the straight line drawn from a given point, and making a given angle with a given straight line, is

yi

b
Nl+be

29 wit

B 50. The perpendiculars from the angles of a triangle on the opposite sides meet in one point

29 51. If the straight line be referred to oblique axes, its equation is

sin. 6
y
x + b

30 ) The tangent of the angle between two given straight lines is tan. (0 - 0) =

(u – a') sin. w

1 + ad + (a + a) cos. W The equation to a straight line making a given angle with a given line is

u sin. w B(1 + a cos. w) Y - Y1 =

(x -x1).

sin. w + B (a + cos.w) The length of the perpendicular from a given point on a given line is

(y1 – AXI 8) sin. w p==

• 31 N{1+2 a cos.wta?}' 52. If upon the sides of a triangle, as diagonals, parallelograms be described,

having their sides parallel to two given lines, the other diagonals of the
parallelograms will intersect each other in the same point

. 31

sin. (w

CHAPTER IV.

THE TRANSFORMATION OF CO-ORDINATES.

32

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53. The object of the transformation of co-ordinates
54. If the origin be changed, and the direction of co-ordinates remain the same,

y=b+ Y, x = a t X
where x and y are the original co-ordinates, X and Y the new ones
55. If the axes be changed from oblique to others also oblique,
X sin. 0 + Y sin. O

1

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sin. w

{ X sin. X Ax +Y sin. Y Ar} sin. « A y
= {Xsin. XA y+Y sin. Y Ay} sin, x Ay

=

X sin.(w—0)+Y sin. (w-ad')

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sin. w

34

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Art.

Page 57. If the original axes be rectangular, and the new oblique,

y=X sin. 8 + Y sin. e',
x = X cos. 8 + Y cos.d.

34 58. Let both systems be rectangular,

y= X sin. 0 + Y cos. 8 = X cos. X Ay + Y cos. Y A y,
x = X cos. 8 * Y sin. 6 = X cos. X A X + Y cos. Y AX.

34 60. To transform an equation between co-ordinates x and y, into another between polar co-ordinates r and 0.

q sin.(0 + ) y=b+

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61. Įf the original axes be rectangular,

y = b + q sin. e,

x = a + cos. 6. 62. To express r and 6 in terms of x and

y,

(y b) sin.w 0 + 0 = tan. -1

3 – a + (y-6) cos.w p= (x – a)2 + (y 5)2 + 2 (z a) (y b) cos. e. 63. If the original axes be rectangular, and the pole at the origin,

y2 + x2 g? = x2 + y2.

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36

CHAPTER V.

ON THE CIRCLE,

37

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64, 65. Let a and 6 be the co-ordinates of the centre, and r the radius, then the equation to the circle referred to rectangular axes is generally

(y b)2 + (x – a)2 = pi. If the origin is at the centre,

y2 + x2 = 72. If the origin is at the extremity of that diameter which is the axis of x,

y2 = 2 rx 22. 66, 67. Examples of Equations referring to Circles 68. Exceptions, when the Locus is a point or imaginary 69. The equation to the straight line touching the circle at a point ac y' is

y y + x2 = r2 or, generally, (y-6)(y' 6) + (x – a) (x' - a) = 42 70. The tangent parallel to a given line, y = ax +b, is

y = ax Ern1 + 62 71. To find the intersection of a straight line and a circle. A straight line cannot

cut a line of the second order in more than two points

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40

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Art.
72. If the axes are oblique, the equation to the circle is

(y – 5)% + (x – a)2 + 2 (y -b) (x – a) cos.w = r2.
Examples.—The equation to the tangent
73, 74. The Polar equation between u and 8 is

2% – 2 cu cos. (6 - «) + 02 - 5% = 0,
or u – 2 {b sin. 6 + a cos. o} u + a2 +6% -9% = 0.

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41

CHAPTER VI.

DISCUSSION OF THE GENERAL EQUATION OF THE SECOND ORDER.

42

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75. The Locus of the equation a ys + bxy + 6x + dy + es +f=0, depends

on the value of 18 4ac. 76. 62 - 4ac negative; the Locus is an Ellipse, a point, or is imaginary, according

as the roots xi and x, of the equation (62 - 4ac) x2 + 2 (bd - 2a e)x+ d 4 a f = 0 are real and unequal, real and equal, or imaginary.

Examples 77. 1% – 4ac positive; the Locus is an Hyperbola if Xi and xq are real and

unequal, or are imaginary; but consists of two straight lines if xı and x, are

real and equal. Examples 78. 62 4ac=0; the Locus is a Parabola when bd 2 ae is real; but if

bd - 2ae = 0, the locus consists of two parallel straight lines, or of one straight line, or is imaginary, according as d? — 4 a f is positive, nothing,

or negative 79. Recapitulation of results

46

48 49

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CHAPTER VII.

REDUCTION OF THE GENERAL EQUATION OF THE SECOND ORDER,

2

80. Reduction of the equation to the form ays + bactyl + call + fl = 0 . 50 81. General notion of a centre of a curve. The ellipse and hyperbola have a centre, whose co-ordinates are

2a e-bd 2cd - be

62 - 4ac' 62-4 ac' 82. Disappearance of the term xy by a transformation of the axes through an angle 6, determined by tan, 20 =

52

m

n=

51

arC

84. The reduced equation is a yll2 + d " + f'= 0, where

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. 61

Art.
86. Definition of the axes
87, 88. The preceding articles when referred to oblique axes
89, 90. Examples of Reduction
91. Reduction of the general equation when belonging to a Parabola
92. Transferring the axes through an angle d, where tan. 20 =
93. The coefficient of x2 or y disappears
94. Transferring the origin reduces the equation to one of the forms,

a'y"? + e'x" = 0, or c'x" ? + d' yo'l
95. Corresponding changes in the situation of the figure
96, 97. The preceding articles when referred to oblique axes
93. Examples of Reduction when the locus is a parabola.

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CHAPTER VIII.

THE ELLIPSE.

.

68

.

.

.

.

100. The equation to the Ellipse referred to the centre and axes is

ao vo + b^r°= ao so 101, 102. Symmetry of the curve with regard to its axes 103. The sq. on MP: the rectangle A M, MA':: sq. on BC:sq.on AC : 104. The ordinate of the Ellipse has to the ordinate of the circumscribing circle

the constant ratio of the axis minor to the axis major 105. A third proportional to the axis major and minor is called the Latus Rectum69 106-108. The Focus ; Eccentricity; Ellipticity :

69 The rectangle AS, SA' = sq. on BC 109. SP = ut ex, HP-a- ex ; SP + HP = A A

70 110. To find the locus of a point P, the sum of whose distances from two fixed points is constant

70 111. The equation to the tangent is a' y y' + b2 x x' = a be

71 113. The equation to the tangent when the curve is referred to another origin

72 114. The rectangle CT, CM = the square on AC; consequently C T is the same for the ellipse and circumscribing circle

73 115. The rectangle C M, MT= the rectangle A M, MA'

73 116. The tangents at the two extremities of a diameter are parallel 117. The equation to the tangent at the extremity of the Latus Rectum is y = a + ex

74 118. The directrix.—The distances of any point from the focus and from the directrix are in the constant ratio of e:1

74 119. The length of the perpendicular from the focus on the tangent,

a text
pa = 12

e al
The rectangle Sy, Hz = the square on BC
120. The locus of y or z is the circle on the axis major
121. The tangent makes equal angles with the focal distances,

62
tan. SPT- = tan. HP Z. Definition of Foci

76 cy

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Art.

Page

122. The length of the perpendicular from the centre on the tangent,

ab P=

Nog 125. If C E is drawn parallel to the tangent, meeting H P in E, then PE = AC 78 126. The equation to the normal

as y
-
x

78
62 x'
a” e 2

62

b 127. CG= ex'; CG' = y'; MG= X'; PG =

62

ao

y – y=(2)

=

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79

79

80

81 82 82 83

.

The rectangle PG, PG' = the rectangle SP, HP 130. All the diameters of the ellipse pass through the centre;

y = ax to c, a?a y + bax = 0, are the chord and corresponding diameter 131. There is an infinite number of pairs of conjugate diameters;

6% tan. & tan. &

a2 133. Equation to the cạrve referred to any conjugate diameters,

aja ya + b12 x? = a, b,? 134. a 2 + b 2 =a2 + 62 135. a, b, sin. (

1) = ab 136. The sq. on QV : the rectangle P V, VP' :: sq. on CD : sq. on CP 137. The ellipse being referred to conjugate axes, the equation to the tangent is

un' yy' + 6,9 x x' = a;% 6,2 138. The ellipse being referred to its axes, the tangent is parallel to the conjugate diameter : the two equations are,

ayy' + 62 x x' = albo, the tangent,

a? yy' + 6*xx' = 0, the parallel conjugate 139. The square upon CD = the rectangle SP, HP 140. The perpendicular from the centre on the tangent,

a b

a? 62 ; or po

al +62-22 141, 142. The product of the tangents of the angles which a pair of supplemental chords makes with the axis major is constant,

62 a a

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143. The tangent of the angle between two supplemental chords, 262 X'y yʻx

2 62 tan. PQP' =

tan. ARA'S a2–52 y2—42'

al-62 144. Supplemental chords are parallel to conjugate diameters 145. The equation to the ellipse, referred to its equal conjugate diameters, is

ye + 2? = ajo 146. The general polar equation,

a? (y' t u sin. 6)2 +62 (x + u cos. 6)2 = al be

al (1

e2) 147. The centre, the pole, u?

e* (cos. 6)

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