CHAPTER X.

THE ELLIPSE CONTINUED.

Diameters.

187. To find the length of a line drawn from any point in a given direction to meet an ellipse.

Let x, y, be the co-ordinates of the point from which the line is drawn; x, y, the co-ordinates of the point to which the line is drawn; @ the inclination of the line to the axis of x;r the length of the line; then (Art. 27)

x=x+r cos 0,

y=y+r sin 0.......... (1).

If (x, y) be on the ellipse these values may be substituted in the equation

a2y2+b2x2 = a3b2; thus

a2 (y′+r sin 0)2 + b2 (x′+r cos 0)2 = a2b2 ;

.. r2 (a2 sin2 0+b2 cos3 0) + 2r (a3y sin 0 +b2x′ cos 0)

+a2y2+b2x2 — a3b2 0.(2).

=

From this quadratic two values of r can be found which are the lengths of the two lines that can be drawn from (x', y') in the given direction to the ellipse.

188. To find the diameter of a given system of parallel chords in an ellipse. (See definition in Art. 148).

Let be the inclination of the chords to the major axis of the ellipse; let x', y', be the co-ordinates of the middle point of any one of the chords; the equation which determines the

lengths of the lines drawn from (x', y') to the curve is (Art. 187)

r2 (a2 sin2 + b2 cos 0)+2r (a'y sin 0+b2x cos 0)

+ a2y2+b2x22 — a2b2 = 0...........

0............(1).

Since (x, y) is the middle point of the chord, the values of r furnished by this quadratic must be equal in magnitude and opposite in sign; hence the coefficient of r must vanish; thus

Considering x and y' as variable, this is the equation to a straight line passing through the origin, that is, through the centre of the ellipse.

Hence every diameter passes through the centre.

Also every straight line passing through the centre is a diameter, that is, bisects some system of parallel chords, for by giving to a suitable value the equation (2) may be made to represent any line passing through the centre.

If ' be the inclination to the axis of x of the diameter which bisects all the chords inclined at an angle ✪ we have from (2)

189. If one diameter bisect all chords parallel to a second diameter, the second diameter will bisect all chords parallel to the first.

1

Let 0, and 0, be the respective inclinations of the two diameters to the major axis of the ellipse. Since the first bisects all the chords parallel to the second, we have

And this is also the only condition that must hold in order that the second may bisect the chords parallel to the first.

190. The tangent at either extremity of any diameter is parallel to the chords which that diameter bisects.

Let h, k, be the co-ordinates of either extremity of a diameter; the inclination to the major axis of the ellipse of the chords which the diameter bisects. Then the values x=h, y=k, must satisfy the equation

But, by Art. 170, the equation to the tangent at (h, k) is

Hence the tangent is parallel to the bisected chords.

191.

DEF.

DEF. Two diameters are called conjugate when each bisects the chords parallel to the other.

From Art. 190 it follows that each of the conjugate diameters is parallel to the tangent at either extremity of the other.

192.

Given the co-ordinates of one extremity of a diameter to find those of either extremity of the conjugate diameter. Let ACA', BCB', be the axes of an ellipse; PCP', DCD', a pair of conjugate diameters.

Let x, y, be the given co-ordinates of P; then the equation to CP is

Since the conjugate diameter DD' is parallel to the tangent at P the equation to DD' is

(2).

In the figure the abscissa of D is negative and that of D' positive; hence the upper sign applies to D and the lower to D'.

The properties of the ellipse connected with conjugate diameters are numerous and important; we shall now give a

few of them.

193. The sum of the squares of two conjugate semi-diameters

is constant.

Let x', y', be the co-ordinates of P; then by the preceding article

Thus the sum of the squares of two conjugate semi-diameters is equal to the sum of the squares of the semi-axes.

194. The area of the parallelogram which touches the ellipse at the ends of conjugate diameters is constant.

Let PCP', DCD', be the conjugate diameters (see Fig. to Art. 192). The area of the parallelogram described so as to touch the ellipse at P, D, P', D', is 4CP. CD sin PCD, or 4p. CD, where p denotes the perpendicular from C on the tangent at P. Let x', y', be the co-ordinates of P; then the equation to the tangent at Pis

Thus the area of any parallelogram which touches the ellipse at the ends of conjugate diameters is equal to the area of the rectangle which touches the ellipse at the ends of the

axes.