the roots of which equation in p are evidently OP, OP', the values of the radius vector answering to any given value of 0 or POC.

Now, by the theory of equations, OP. OP', the product of these roots will = d2 — r2, a quantity independent of 0, and therefore constant, whatever be the direction in which the line OP is drawn. If the point be outside the circle, it is plain that d2 - 2 must be the square of the tangent.

་

Ex. 2. If through a fixed point O any chord of a circle be drawn, and OQ taken an arithmetic mean between the segments OP, OP', to find the locus of Q.

We have OP+ OP', or the sum of the roots of the quadratic in the last example, = 2d cos 0; but OP + OP′ = 20Q, therefore

The question in this example might have been otherwise stated: "To find the locus of the middle points of chords which all pass through a fixed point."

Ex. 3. If the line OQ had been taken a harmonic mean between OP and OP' to find the locus of Q.

20P.OP'
OP+OP"

That is to say, 0Q= but OP.OP'd2-r2, and OP+ OP' = 2d cos 0; therefore the polar equation of the locus is

This is the equation of a right line (Art. 44) perpendicular to OC, and at a

We can, in like manner, solve this and similar questions when the equation is given in the form

a (x2 + y2) + 2gx + 2fy + c = 0,

for, transforming to polar coordinates, the equation becomes

and, proceeding precisely as in this example, we find, for the locus of harmonic means,

and, returning to rectangular coordinates, the equation of the locus is

gx+fy + c = 0,

the same as the equation of the polar obtained already (Art. 89).

Ex. 4. Given a point and a right line or circle; if on OP the radius vector to the line or circle a part 0Q be taken inversely as OP, find the locus of Q.

Ex. 5. Given vertex and vertical angle of a triangle and rectangle under sides, if one extremity of the base describe a right line or a circle, find the locus described by the other extremity.

Take the vertex for pole; let the lengths of the sides be p and p', and the angles they make with the axis ✪ and e', then we have pp' = k2 and ✪ 0' = C.

The student must write down the polar equation of the locus which one base angle is said to describe; this will give him a relation between p and ; then, writing for p,

k2

and for 0, C+0', he will find a relation between p' and ', which will be the polar equation of the locus described by the other base angle.

This example might be solved in like manner, if the ratio of the sides, instead of their rectangle, had been given.

Ex. 6. Through the intersection of two circles a right line is drawn; find the locus of the middle point of the portion intercepted between the circles.

The equations of the circles will be of the form

Ex. 7. If through any point 0, on the circumference of a circle, any three chords be drawn, and on each, as diameter, a circle be described, these three circles (which, of course, all pass through 0) will intersect in three other points, which lie in one right line (See Cambridge Mathematical Journal, vol. 1. p. 169).

Take the fixed point O for pole, then if d be the diameter of the original circle, its polar equation will be (Art. 95)

p = d cos 0.

In like manner, if the diameter of one of the other circles make an angle a with the fixed axis, its length will be = d cos a, and the equation of this circle will be

p = d cos a cos (0 — a).

The equation of another circle will, in like manner, be

p = d cosẞ cos (0 – ẞ).

To find the polar coordinates of the point of intersection of these two, we should seek what value of § would render

and it is easy to find that must = a+ß, and the corresponding value of p = d cos a cos B.

Similarly, the polar coordinates of the intersection of the first and third circles are

0 = a+y, and p = d cos a cos y.

Now, to find the polar equation of the line joining these two points, take the general equation of a right line, p cos (k - 0) = p (Art. 44), and substitute in it successively these values of 0 and p, and we shall get two equations to determine p and k. We shall get

Hence

p

= d cos a cosẞ cos {k — (a + ẞ)} = d cos a cos y cos {k - (a + y)}.

k = a+B+y; and pd cos a cosẞ cos y.

The symmetry of these values shows that it is the same right line which joins the intersections of the first and second, and of the second and third circles, and, therefore, that the three points are in a right line.

CHAPTER VIII.

PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES.

105. To find the equation of the chord of intersection of two circles.

If S=0, S'=0 be the equations of two circles, then any equation of the form 8+k8'0 will be the equation of a figure passing through their points of intersection (Art. 40).

Let us write down the equations

S = (x − a )2 + (y − B )2 — p2 = 0,

S' = (x − α' ́)3 + (y — B′)2 — p22 = 0,

and it is evident that the equation S+kS'=0 will in general represent a circle, since the coefficient of xy=0, and that of x2 = that of y3. There is one case, however, where it will represent a right line, namely, when k=-1. The terms of the second degree then vanish, and the equation becomes

12

S – S'= 2 (α′ – a) x + 2 (B′ − ẞ) y + 222 — p2 + a2 — a22 + ß2 — ß”2 = 0. +a2-a^" This is, therefore, the equation of the right line passing through the points of intersection of the two circles.

What has been proved in this article may be stated as in Art. 50. If the equation of a circle be of the form S+kS' = 0 involving an indeterminate k in the first degree, the circle passes through two fixed points, namely, the two points common to the circles S and S'.

106. The points common to the circles S and S′ are found by seeking, as in Art. 82, the points in which the line S-S meets either of the given circles. These points will be real, coincident, or imaginary, according to the nature of the roots of the resulting equation; but it is remarkable that, whether the circles meet in real or imaginary points, the equation of the chord of intersection, S-S=0, always represents a real line, having important geometrical properties in relation to the two circles. This is in conformity with our assertion (Art. 82), that

the line joining two points may preserve its existence and its properties when these points have become imaginary.

In order to avoid the harshness of calling the line S- S', the chord of intersection in the case where the circles do not geometrically appear to intersect, it has been called* the radical axis of the two circles.

107. We saw (Art. 90) that if the coordinates of any point xy be substituted in S, it represents the square of the tangent drawn to the circle S from the point xy. So also S' is the square of the tangent drawn to the circle S'; hence the equation S - S′ =0 asserts, that if from any point on the radical axis tangents be drawn to the two circles, these tangents will be equal.

The line (S– S′) possesses this property whether the circles meet in real points or not. When the circles do not meet in real points, the position of the radical axis is determined geometrically by cutting the line joining their centres, so that the difference of the squares of the parts may the difference of the squares of the radii, and erecting a perpendicular at this point; as is evident, since the tangents from this point must be equal to each other.

If it were required to find the locus of a point whence tangents to two circles have a given ratio, it appears, from Art. 90, that the equation of the locus will be S-k3S'=0, which (Art. 105) represents a circle passing through the real or imaginary points of intersection of S and S'. When the circles S and S' do not intersect in real points, we may express the relation which they bear to the circle S-S', by saying that the three circles have a common radical axis.

Ex. Find the coordinates of the centre, and the radius of kS + IS'.

Ans. Coordinates are

ka + la' kB + lẞ; that is to say, the line joining the centres

k + l'

of S, S' is divided in the ratio kl.

k + l

;

Radius is given by the equation (k + 1)2 p112 = (k + 1) (kr2 + lr12) — klD2,

where D is the distance between the centres of S and S'.

108. Given any three circles, if we take the radical axis of each pair of circles, these three lines will meet in a point, which is called the radical centre of the three circles.

* By M. Gaultier, of Tours (Journal de l'École Polytechnique, Cahier xvI. 1813).

For the equations of the three radical axes are

S- S'=0, S′- S′′ =0, S′′ – S=0,

which, by Art. 41, meet in a point.

From this theorem we immediately derive the following: If several circles pass through two fixed points, their chords of intersection with a fixed circle will pass through a fixed point.

For, imagine one circle through the two given points to be fixed, then its chord of intersection with the given circle will be fixed; and its chord of intersection with any variable circle drawn through the given points will plainly be the fixed line joining the two given points. These two lines determine by their intersection a fixed point through which the chord of intersection of the variable circle with the first given circle must pass.

Ex. 1. Find the radical axis of

x2 + y2 - 4x — 5y +7=0; x2 + y2 + 6x + 8y — 9 = 0.

Ex. 2. Find the radical centre of

Ans. 10x+13y = 16.

(x − 1)2 + (y − 2)2 = 7; (x − 3)2 + y2 = 5; (x + 4)2 + (y + 1)2 = 9.

Ans. (-1, -25).

*109. A system of circles having a common radical axis possesses many remarkable properties, which are more easily investigated by taking the radical axis for the axis of Y1 and the line joining the centres for the axis of x. Then the equation of any circle will be

x2 + y2 - 2kx ± S2 = 0,

x=0

where S2 is the same for all the circles of the system, and the equations of the different circles are obtained by giving different values to k. For it is evident (Art. 80) that the centre is on the axis of x, at the variable distance k; and if we make x= in the equation, we see that no matter what the value of k may be, the circle passes through the fixed points on the axis of y, y2+8=0. These points are imaginary when we give & the sign +, and real when we give it the sign –

*110. The polars of a given point, with regard to a system of circles having a common radical axis, always pass through a fixed point.

The equation of the polar of x'y' with regard to x2 + y2 — 2kx+82 = 0,