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) = b22.

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

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

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.

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 F and 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

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

major at a distance from the centre = ±

a2

с

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

=

But the distance of any point on the curve from the focus a-ex'. 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 (Ax + 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

FT = /, (a — ex') = /, FP.

b

Ε' Τ' =

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

b

makes with the tangent. 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.

P

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.

F

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

at that point will be at right angles to the tangent to the

hyperbola.

Ex. 1. Prove analytically that confocal conics cut at right angles.

The coordinates of the intersection of the conics

satisfy the relation obtained by subtracting the equations one from the other, viz.

But if the conics be confocal, a2 — a'2 = b2 — 6'2, and this relation becomes

Ex. 2. Find the length of a line drawn through the centre parallel to either focal radius vector, and terminated by the tangent.

This length is found by dividing the perpendicular from the centre on the tangent the sine of the angle between the radius vector and tangent, and is

(ab), by (1)

therefore = a.

Ex. 3. Verify that the normal, which is a bisector of the angle between the focal radii, divides the distance between the foci into parts which are proportional to the focal radii (Euc. VI. 3). The distance of the foot of the normal from the centre is (Art. 180) = e2x'. Hence its distances from the foci are c + e2x' and ce2x', quantities which are evidently e times a + ex' and a — - ex'.

Ex. 4. To draw a normal to the ellipse from any point on the axis minor.

Ans. The circle through the given point and the two foci, will meet the curve at the point whence the normal is to be drawn.

189. Another important consequence may be deduced from the theorem of Art. 187, that the rectangle under the focal perpendiculars on the tangent is constant.

For, if we take any two tangents, we have (see figure, next

page)

FT

but is the ratio of the sines of the parts into which the line

Ft

F't

FT

FP divides the angle at P, and is the ratio of the sines of

the parts into which F'P divides the same angle; we have, therefore, the angle TPF=ť'PF'.

If we conceive a conic section to pass through P, having F and F for foci, it was proved in Art. 188, that the tangent to it must be equally inclined to the lines FP, F'P; it follows, therefore, from the present Article, that it must be also

P

T

T

F'

equally inclined to PT, Pt; hence we learn that if through any point (P) of a conic section we draw tangents (PT, Pt) to a confocal conic section, these tangents will be equally inclined to the tangent at P.

190. To find the locus of the foot of the perpendicular let fall from either focus on the tangent.

The perpendicular from the focus is expressed in terms of the angles it makes with the axis by putting x = c, y' = 0 in the formula of Art. 178, viz.,

p = √(a2 cos2 a + b2 sin2 a) — x cos a — y' sin a.

Hence the polar equation of the locus is

or

or

=√(a2 cos" a+b2 sin2 a) - c cosa,

p2+2cp cos a + c2 cos2 a = a2 cos" a+b2 sin2 a,
p2+2cp cos a=b*.

=

This (Art. 95) is the polar equation of a circle whose centre is on the axis of x, at a distance from the focus -c; the circle is, therefore, concentric with the curve. The radius of the circle is, by the same Article, = a.

Hence, If we describe a circle having for diameter the transverse axis of an ellipse or hyperbola, the perpendicular from the focus will meet the tangent on the circumference of this circle.

Or, conversely, if from any point F (see figure, p. 177) we draw a radius vector FT to a given circle, and draw TP perpendicular to FT, the line TP will always touch a conic section, having F for its focus, which will be an ellipse or hyperbola, according as F is within or without the circle.

It may be inferred from Art. 188, Ex. 2, that the line CT, whose length = a, is parallel to the focal radius vector F'P.