If the co-ordinates of any point P be (x', y'), the equation of the line through P perpendicular to its polar with respect to the conic will be

If this line pass through the point (h, k), we have

From (i) we see that (x, y) is on a rectangular hyperbola ...... (a). The equation of the circle circumscribing the triangle cut off from the axes by the polar of (x', y') will be

The circle will pass through the point (Ah, Ak) if

Hence, if (x, y) satisfies the relation (i), we have

a2 - b2
λ=
h2 + k2°

Hence the circles all pass through the point O' whose co-ordinates are

The point O' is on the circle circumscribing the triangle formed by the axes and any one of the polars; hence the parabola whose focus is O' and which touches the axes will touch every one of the polars............(7).

The parabola touches the axes of the original conic, therefore the centre C is a point on the directrix of the parabola. Also the lines CO and CO' make equal angles with the axis of x, which is a tangent to the parabola; therefore O' being the focus, CO is the directrix.......... (§).

Since CO'. CO=a2-b2, and CO, CO make equal angles with the axis of x, and are on the same side of the axis of y, the points O and Q' are interchangeable ........(€).

199. Definition. Two curves are said to be similar and similarly situated when radii vectores drawn to the first from a certain point O are in a constant ratio to

parallel radii vectores drawn to the second from another point O'.

Two curves are similar when radii drawn from two fixed points O and O' making a constant angle with one another are proportional.

The two fixed points O and O' may be called centres of similarity.

200. If one pair of centres of similarity exist for two curves, then there will be an infinite number of such pairs.

Let O, O' be the given centres of similarity, and let OP, O'P' be any pair of parallel radii. Take C any point whatever, and draw O'C' parallel to OC and in the ratio O'P OP. Then, from the similar triangles COP, and COP' we see that CP is parallel to C'P' and in a constant ratio to it; which proves that C, C' are centres of similarity.

201. If two central conics be similar the centres of the two curves will be centres of similarity.

Let 0 and O' be two centres of similarity. Draw any chord POQ of the one, and the corresponding chord PO'Q of the other. Then by supposition PO. OQ:P'O'.O'Q is constant for every pair of corresponding chords. But since is a fixed point PO. OQ is always in a constant ratio to the square of the diameter of the first conic which is parallel to it. The same applies to the other conic. Therefore corresponding diameters of the two conics are in a constant ratio to one another; this shews that the centres of the curves are centres of similarity.

202. To find the conditions that two conics may be similar and similarly situated.

By the preceding Article, their respective centres are centres of similarity.

Let the equations of the conics referred to those centres and parallel axes be

and

ax2 + 2hxy + by2 + c = 0, a2x2+2hxy+b'y2 + c′ = 0;

or, in polar co-ordinates,

r2 (a cos2 0 + 2h sin ◊ cos 0 + b sin3 0) + c = 0, and r12 {a' cos2 0 + 2h′ sin 0 cos 0 + b′ sin3 0} + c′ = 0.

12

If therefore r2: r" be constant, we must have

of the two conics are parallel. [This result may be obtained in the following manner; since r: r' is constant, when one of the two becomes infinite, the other will also be infinite, which shews that the asymptotes are parallel.]

Conversely, if these conditions be satisfied, and if each fraction be equal to λ, then

therefore the ratio of corresponding radii is constant, and therefore the curves are similar.

If c and Ac' have not the same sign the constant ratio is imaginary, and is zero or infinite if c or c be zero.

The conditions of similarity are satisfied by the three curves whose equations are

xy = c, xy= 0, and xy: =-C.

Therefore an hyperbola, the conjugate hyperbola, and their asymptotes are three similar and similarly situated curves; the constant ratio being 1 for the conjugate hyperbola, and zero for the straight lines.

These curves have not however the same shape. For similar curves to have the same shape the constant ratio must be real and finite.

203. To find the condition that two conics may be similar although not similarly situated.

We have seen that the centres of the two curves must be centres of similarity.

Let the equations of the curves referred to their respective centres be

ax2 + 2hxy + by2+ c = 0

a'x2+2h'xy + b'y2 + c′ = 0 ....

and let the chord which makes an angle

(i),

.(ii),

with the axis of

x in the first be proportional, for all values of 0, to that which makes an angle (0+ a) in the second. If the axes of the second conic be turned through the angle a, we shall then have radii of the two conics which make the same angle with the respective axes in a constant ratio.

Let the equation of the second conic become
A'x2+ 2H'xy + B'y2 + c′ = 0,

Then, by the preceding Article, we must have
A' H' B
a = h = b;

therefore

=

A' + B′ __ √(A'B' — H′′2)
a+b √(ab — h2)

But [Art. 52] A'+ B'=a'+b', and A'B'— H'2== a'b' — h2'; therefore the condition of similarity is

The above shews that the angles between the asymptotes of similar conics are equal. [See Art. 174.]

This result may also be obtained in the following manner: since radii vectores of the two curves which are inclined to one another at a certain constant angle are in a constant ratio, it follows that the angle between the two directions which give infinite values for the one curve must be equal to the corresponding angle for the other, that is to say the angle between the asymptotes of the one conic is equal to the angle between the asymptotes of the other.

EXAMPLES ON CHAPTER X.

1. If Q and P be any two points, and C the centre of a conic; shew that the perpendiculars from Q and C on the polar of P with respect to the conic, are to one another in the same ratio as the perpendiculars from P and C on the polar of Q.

2. Two tangents drawn to a conic from any point are in the same ratio as the corresponding normals.

3. Find the loci of the fixed points of the examples in Article 195, for different positions of O on the conic.

4. POQ is one of a system of parallel chords of an ellipse, and O is the point on it such that PO+ OQ is constant; shew that, for different positions of the chord, the locus of O is a concentric conic.

5. If O be any fixed point and OPP' any chord cutting a conic in P, P', and on this line a point D be taken such that

the locus of D will be a conic whose centre

6. If OPP'QQ' is one of a system of parallel straight lines cutting one given conic in P, P' and another in Q, Q', and O is such that the ratio of the rectangles OP. OP' and OQ. OQ' is constant; shew that the locus of Ō is a conic through the intersections of the original conics.

7. POP', QOQ' are any two chords of a conic at right angles to one another through a fixed point 0; shew that 1

PO.OP

1

QO.OQ'

is constant.

8. If a point be taken on the axis-major of an ellipse,

be prove that the sum of

a2 +

the squares of the reciprocals of the segments of any chord

passing through that point is constant.