or, since (x', y') and (x”, y′′) are both on the curve,

.(i),

· 1 ........ .........(ii),

y" = ±-x'√− 1

a

.(iii).

So that, as in the case of the ellipse, the sum of the squares of two conjugate diameters is constant.

146. Definition. An asymptote is a straight line which meets a curve in two points at infinity, but which is not altogether at infinity.

To find the asymptotes of an hyperbola.

To find the abscissæ of the points where the straight line y = mx +c cuts the hyperbola, we have the equation

Both roots of the equation (i) will be infinite if the

coefficients of x and of x are both zero; that is, if

b

y=±x; or, expressed in one equation,

a

Draw lines through B, B' parallel to the transverse axis, and through A, A' parallel to the conjugate axis; then we see from (ii) that the asymptotes are the diagonals of the rectangle so formed.

The ellipse has no real points at infinity, and therefore the asymptotes of an ellipse are imaginary.

147. Any straight line parallel to an asymptote will meet the curve in one point at infinity.

For, one root of the equation (i) Art. 146 will be infinite, if the coefficient of x2 is zero. This will be the case

b

b

if m = ± So that the line y = ± x+c meets the

α

curve in one point at infinity whatever the value of c may be.

148. The equation of the hyperbola which has BB' for its transverse axis and AA' for its conjugate axis is

This hyperbola and the original hyperbola, whose equation is

x2 y2
= 1
a2 b2

are said to be conjugate to one another.

.....

(ii),

144

53. If a length PQ be taken in the normal at any point P of an ellipse whose centre is C, equal in length to the semidiameter which is conjugate to CP, shew that hes on one or other of two circles.

54. Shew that, if & be the angle between the tangents to the ellipse + -1=0 drawn from the point (”, y^), then

a2 b2

55. TP, TQ are the tangents drawn from an external point (x, y) to the ellipse

+ -1=0; shew that, if S be a b2

focus,

ST2
SP.SQ

56. If two tangents to an ellipse from a point 7 intersect at an angle o, shew that ST. HT cos &=CT12 — ď2 — b2, C is the centre of the ellipse and S, H the foci.

where

57. If the perpendicular from the centre C of an ellipse on the tangent at any point P meet the focal distance SP, produced if necessary, in R; the locus of R will be a circle.

58. If two concentric ellipses be such that the foci of one lie on the other, and if e, e' be their eccentricities, shew that their axes are inclined at an angle cos € + e −1

59. Shew that the angle which a diameter of an ellipse subtends at either end of the axis-major is supplementary to that which the conjugate diameter subtends at the end of the axis-minor.

60. If 6, be the angles subtended by the axis majo an ellipse at the extremities of a pair of conjugate diar shew that cot + cot' is constant,

61. If the distance between the foci of angles 26, 26' at the ends of a pair of conj. that tan2+ tan 6' is constant.

62. If λ, A' be the ages which any two con u. ters subtend at any fit on an ele cotλ+cotλ' is constant.

63. Shew that pairs of comme Camere are cut in involution by any strain ine

64. A triangle whose sice tove

it, is a minimum; shew text

C

at its middle point, and that the triari. "Jers points of contact is a maximu

65. A, B, C, D are our ne poma

any other point on the surv perpendiculars from ca. the product of the perpetuats

66. Find the los normals to an einpse Wind

07. Find the equat

section of the tangent with the normal het

154

Eher mey the asumatates is its ven at diameter.

THE HYPERBOLA.

We append some properties of a pair of conjugate hyperbolas.

(1) The two hyperbolas have the same asymptotes. (2) If two diameters be conjugate with respect to one of the hyperbolas, they will be conjugate with respect to the other.

This follows from the condition in (ix) Article 143.

(3) The equations of the hyperbolas (ii) and (i) can [Art. 142] be written in the forms

It is clear that if, for any value of 0, r2 is positive for one curve it is negative for the other.

Hence every diameter meets one curve in real points and the other in imaginary points; moreover the lengths of semi-diameters of the two curves are, for all values of 0, connected by the relation r-r. Time

1

2

2

2

(4) If two conjugate diameters cut the curves (ii) and (i) in P and d respectively, then CP2 - Cd a2-b2.

=

Let x', y' be the co-ordinates of P, and x", y" the co-ordinates of d.