Substituting in the second the values of x and y found from the first equation, and remembering that x, y satisfy the equation

of the curve, we find without difficulty

173. To express the lengths of a diameter (a'), and its conjugate (b'), in terms of the abscissa of the extremity of the diameter. (1) We have

But

Hence

a" = x2+y".

or, The sum of the squares of any pair of conjugate diameters of an ellipse is constant (see Ex. 3, Art. 159).

174. In the hyperbola we must change the signs of b3 and b", and we get

a” —b" — a" — b3,

or, The difference of the squares of any pair of conjugate diameters of a hyperbola is constant.

If in the hyperbola we have a=b, its equation becomes

x2 - y2 = a2,

and it is called an equilateral hyperbola.

The theorem just proved shows that every diameter of an equilateral hyperbola is equal to its conjugate.

The asymptotes of the equilateral hyperbola being given by the equation

x3 — y2 = 0,

are at right angles to each other. Hence this hyperbola is often called a rectangular hyperbola.

The condition that the general equation of the second degree should represent an equilateral hyperbola is a =-b; for (Art. 74) this is the condition that the asymptotes (ax + 2hxy + by*) should be at right angles to each other; but if the hyperbola be re tangular it must be equilateral, since (Art. 167) the tangent of half the angle between the asymptotes

b

=

a

; therefore, if

175. To find the length of the perpendicular from the centre on the tangent.

The length of the perpendicular from the origin on the line

176. To find the angle between any pair of conjugate dia

meters.

The angle between the diameters is equal to the angle be tween either, and the tangent parallel to

T

P

the other.

Now

p'

The equation a'b' sinab proves that the triangle formed by joining the extremities of conjugate diameters of an ellipse or hyperbola has a constant area (see Art. 159, Ex. 2).

177. The sum of the squares of any two conjugate diameters of an ellipse being constant, their rectangle is a maximum when they are equal; and, therefore, in this case, sin is a minimum; hence the acute angle between the two equal conjugate diameters is less (and, consequently, the obtuse angle greater) than the angle between any other pair of conjugate diameters.

The length of the equal conjugate diameters is found by making a b' in the equation a" + b2 = a2 + b2, whence a"" is half the sum of a and b, and in this case

The angle which either of the equi-conjugate diameters makes with the axis of x is found from the equation

by making tan- tane'; for any two equal diameters make equal angles with the axis of x on opposite sides of it (Art. 162).

It follows, therefore, from Art. 167, that if an ellipse and hyperbola have the same axes in magnitude and position, then the asymptotes of the hyperbola will coincide with the equi-conjugate diameters of the ellipse.

The general equation of an ellipse, referred to two conjugate diameters (Art. 168), becomes x+y=a", when a'=b'. We see, therefore, that, by taking the equi-conjugate diameters for axes, the equation of any ellipse may be put into the same form as the equation of the circle, x2+ y2 = r2, but that in the case of the ellipse the angle between these axes will be oblique.

178. To express the perpendicular from the centre on the tangent in terms of the angles which it makes with the axes. If we proceed to throw the equation of the tangent [xx yy 1) into the form x cosa+y sina=p (Art. 23),

+

a2

b2

=

Substituting in the equation of the curve the values of x', y', hence obtained, we find

p2 = a' cos'a + b2 sin' a.*

The equation of the tangent may, therefore, be written

x cosa + y sina - √√(a2 cos3a + b2 sin2a) = 0.

Hence, by Art. 34, the perpendicular from any point (x'y') on the tangent is

√(a' cos'a + b2 sin' a) - x' cosa - y' sina,

where we have written the formula so that the perpendiculars shall be positive when x'y' is on the same side of the tangent

as the centre.

Ex. To find the locus of the intersection of tangents which cut at right angles. Let p, p' be the perpendiculars on those tangents, then

p2 = a2 cos2a + b2 sin2a, p22 = a2 sin2a + b2 cos2a, p2 + p′′2 = a2 + b2.

But the square of the distance from the centre, of the intersection of two lines which cut at right angles, is equal to the sum of the squares of its distances from the lines themselves. The distance, therefore, is constant, and the required locus is a circle (see p. 166, Ex. 4).

179. The chords which join the extremities of any diameter to any point on the curve are called supplemental chords. Diameters parallel to any pair of supplemental chords are conjugate.

For if we consider the triangle formed by joining the extremities of any diameter AB to any point on the curve D; since, by elementary geometry, the line joining the middle points of two sides must be parallel to the third, the diameter bisecting AD will be parallel to BD, and the diameter bisecting BD will be parallel to AD. The same thing may be proved analytically, by forming the equations of AD and BD, and showing that the product of the tangents of the angles made by these lines with b2 the axis is a2.

This property enables us to draw geometrically a pair of conjugate diameters making any angle with each other. For if we describe on any diameter a segment of a circle, containing the

* In like manner, p2 = a" cos2a + b2 cos2ß, a and ẞ being the angles the perpendicular makes with any pair of conjugate diameters.

given angle, and join the points where it meets the curve to the extremities of the assumed diameter, we obtain a pair of supplemental chords inclined at the given angle, the diameters parallel to which will be conjugate to each other.

Ex. 1. Tangents at the extremities of any diameter are parallel.

This also follows from the first theorem of Art. 146, and from considering that the centre is the pole of the line at infinity (Art. 154).

Ex. 2. If any variable tangent to a central conic section meet two fixed parallel tangents, it will intercept portions on them, whose rectangle is constant, and equal to the square of the semi-diameter parallel to them.

Let us take for axes the diameter parallel to the tangents and its conjugate, then the equations of the curve and of the variable tangent will be

The intercepts on the fixed tangents are found by making ≈ alternately = ±a' in the latter equation, and we get

b'2 y=

F

which, substituting for y" from the equation of the curve, reduces to b”.

Ex. 3. The same construction remaining, the rectangle under the segments of the variable tangent is equal to the square of the semi-diameter parallel to it.

For, the intercept on either of the parallel tangents is to the adjacent segment of the variable tangent as the parallel semi-diameters (Art. 149); therefore, the rectangle under the intercepts of the fixed tangent is to the rectangle under the segments of the variable tangent as the squares of these semi-diameters; and, since the first rectangle is equal to the square of the semi-diameter parallel to it, the second rectangle must be equal to the square of the semi-diameter parallel to it.

Ex. 4. If any tangent meet any two conjugate diameters, the rectangle under its segments is equal to the square of the parallel semi-diameter.

Take for axes the semi-diameter parallel to the tangent and its conjugate; then the equations of any two conjugate diameters being (Art, 170)

y' xx' yy'
y= x,
+ = 0,
a'2
6'2

the intercepts made by them on the tangent are found, by making x = a', to be

We might, in like manner, have given a purely algebraical proof of Ex. 3. Hence, also, if the centre be joined to the points where two parallel tangents meet any tangent, the joining lines will be conjugate diameters.

Ex. 5. Given, in magnitude and position, two conjugate semi-diameters, Oa, Ob, of a central conic, to determine the position of the axes.