where X, Y are the coordinates of the intersection of tangents, found in the last Example.

The values of X and Y may be written in other forms. Since by combining the equations

+

x'2 y'2

112

1,

a2

a2

+2 = 1,

181 (a). Let CP, CQ be a pair of conjugate semi-diameters

of an ellipse; let the normal PN meet CQ in R; take PD,

D

Hence CD'2 = (a+b). Similarly for CD. The axis-major bisects the angle DCD'.

For the line

b

Similarly DN== (a - b). At the point N, therefore, the

α

base of the triangle DCD' is divided in the ratio of the sides, and, therefore, CN is the internal bisector of the vertical angle. In like manner, it is proved that CN' is the external bisector.

Hence then, being given two conjugate semi-diameters CP, CQ in magnitude and position, we are given the axes in magnitude and position. For we have only from P to let fall on CQ the perpendicular PR; to take PD, PD' each equal CQ; then the axes are in direction the bisectors of the angle DCD'; while their lengths are the sum and difference of CD, CD'.

THE FOCI.

182. If on the axis major of an ellipse we take two points

equidistant from the centre whose common distance

=±√√(a2-b2), or =±c,

these points are called the foci of the

curve.

T'

R

P

T

F'

CN F

The foci of a hyperbola are two points on the transverse axis, at a distance from the centre still =+c, c being in the hyperbola

= √√(a2+b2).

To express the distance of any point on an ellipse from the focus.

Since the coordinates of one focus are (x=+c, y=0), the square of the distance of any point from it

[We reject the value (ex′ – a) obtained by giving the other sign to the square root. For, since a is less than a, and e less than 1, the quantity ex-a is constantly negative, and therefore does not concern us, as we are now considering, not the direction, but the absolute magnitude of the radius vector FP.] We have, similarly, the distance from the other focus

F'P=a+ex',

since we have only to write - c for + c in the preceding formula. Hence FP+F'P=2a,

or,

The sum of the distances of any point on an ellipse from the foci is constant, and equal to the axis major.

183. In applying the preceding proposition to the hyperbola, we obtain the same value for FP"; but in extracting the square

A A

root we must change the sign in the value of FP, for in the hyperbola a' is greater than a and e is greater than 1. Hence, a-ex' is constantly negative; the absolute magnitude therefore of the radius vector is

Therefore, in the hyperbola, the difference of the focal radii is constant, and equal to the transverse axis.

The rectangle under the focal radii =(a*- e2x2), that is, (Art. 173)=6′′.

184. The reader may prove the converse of the above results by seeking the locus of the vertex of a triangle, if the base and either sum or difference of sides be given.

Taking the middle point of the base (=2c) for origin, the equation is

√{y3 + (c+x)2} ± √√ {y2 + (c − x)2} = 2a,

which, when cleared of radicals, becomes

Now, if the sum of the sides be given, since the sum must always be greater than the base, a is greater than c, therefore the coefficient of y' is positive, and the locus an ellipse.

2

If the difference be given, a is less than c, the coefficient of y* is negative, and the locus a hyperbola.

185. By the help of the preceding theorems we can describe an ellipse or hyperbola mechanically.

If the extremities of a thread be fastened at two fixed points Fand F", it is plain that a pencil moved about so as to keep the thread always stretched will describe an ellipse whose foci are F and F, and whose axis major is equal to the length of the thread. In order to describe a hyperbola, let a ruler be fastened at one extremity (F), and capable of moving round it, then if a thread, fastened to a fixed point F', and also to a fixed point on the ruler (R), be kept stretched by a ring at P, as the ruler is moved round, the point

F'

R

P will describe a hyperbola; for, since the sum of F'P and PR is constant, the difference of FP and F'P will be constant.

186. The polar of either focus is called the directrix of the conic section. The directrix must, therefore

(Art. 169), be a line perpendicular to the axis

a

major at a distance from the centre = ± C

Knowing the distance of the directrix from the centre, we can find its distance from any point on the curve. It must be equal to

= a— ex'.

α

с

But the distance of any point on the curve from the focus Hence we obtain the important property, that the distance of any point on the curve from the focus is in a constant ratio to its distance from the directrix, viz. as e to 1.

Conversely, a conic section may be defined as the locus of a point whose distance from a fixed point (the focus) is in a constant ratio to its distance from a fixed line (the directrix). On this definition several writers have based the theory of conic sections. Taking the fixed line for the axis of x, the equation of the locus is at once written down

which it is easy to see will represent an ellipse, hyperbola, or parabola, according as e is less, greater than, or equal to 1.

Ex. If a curve be such that the distance of any point of it from a fixed point. can be expressed as a rational function of the first degree of its coordinates, then the curve must be a conic section, and the fixed point its focus (see O'Brien's Coordinate Geometry, p. 85).

For, if the distance can be expressed

p = Ax+ By + C,

since Ax + By + C is proportional to the perpendicular let fall on the right line whose equation is (4x + By + C = 0) the equation signifies that the distance of any point of the curve from the fixed point is in a constant ratio to its distance from this line.

187. To find the length of the perpendicular from the focus on the tangent.

The length of the perpendicular from the focus (+c, 0) on

Hence

α

b

4

ab

b

b

b

b

FT.F′′ T′ = b2 (since a2 — e2x22 = b′′2),

or, The rectangle under the focal perpendiculars on the tangent is constant, and equal to the square of the semi-axis minor.

This property applies equally to the ellipse and the hyperbola.

188. The focal radii make equal angles with the tangent.

Hence the sine of the angle which the focal radius vector FP

makes with the tangent

b

But we find, in like manner,

the same value for sin F'PT", the sine of the angle which the

other focal radius vector F'P makes with the tangent.

The theorem of this article is true both for the ellipse and hyperbola, and, on looking at the figures, it is evident that the tangent to the ellipse is the external bisector of the angle between the focal radii, and the tangent to the hyperbola the internal bisector.

T

Hence, if an ellipse and hyperbola, having the same foci, pass through the same point, they will cut each other at right angles, that is to say, the tangent to the ellipse