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the chord be 2r sind, then 0′-0′′ 28. The co-ordinates therefore found in the last example fulfil the condition

(x2 + y2) cos2 = r2.

Ex. 3. What is the locus of a point where a chord of a constant length is cut in a given ratio?

Writing down (Art. 7) the co-ordinates of the point where the chord is cut in a given ratio, it will be found that they satisfy the condition x2 + y2 = constant.

103. We have seen that the tangent to any circle x2+ y2= r2 has an equation of the form

x cose+y sin0=r;

and it can be proved, in like manner, that the equation of the tangent to (x − a)2 + (y − ẞ)2 = r2 may be written

-

(x-a) cose+(y-B) sine = r.

Conversely, then, if the equation of any right line contain an indeterminate in the form

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that line will touch the circle (x − a)2 + (y — ß)2 = r3.

Ex. 1. If a chord of a constant length be inscribed in a circle, it will always touch another circle. For, in the equation of the chord

x cos 1⁄2 (0′+0′′) + y sin 1 (0′ + 0′′) =
=rcos (0′ – 0′′) ;

by the last article, 0' — 0′′ is known, and 0′ + 0′′ indeterminate; the chord, therefore, always touches the circle

x2 + y2 = r2 cos2d.

Ex. 2. Given any number of points, if a right line be such that m' times the perpendicular on it from the first point, +m" times the perpendicular from the second, + &c., be constant, the line will always touch a circle.

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This only differs from Ex. 4, p. 49, in that the sum, in place of being = 0, is constant. Adopting then the notation of that Article, instead of the equation there found, {x (m) - (mx')} cos a + {yΣ (m) — Σ (my')} sin a = = 0,

we have only to write

{xΣm - Σ (mx')} cos a + {yΣ (m) — Σ (my')} sin a = constant.

Hence this line must always touch the circle

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2

= = constant,

Σ (m) }

whose centre is the centre of mean position

Σ (m)

of the given points.

104. We shall conclude this Chapter with some examples of

the use of polar co-ordinates.

Ex. 1. If through a fixed point any chord of a circle be drawn, the rectangle under its segments will be constant (Euclid III. 35, 36).

Take the fixed point for the pole, and the polar equation is (Art. 95)

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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 O be outside the circle, it is plain that d2r2 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' = 202, therefore

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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, OQ= but OP.OP'd2 - r2, and OP+OP′ = 2d cos 0; therefore the polar equation of the locus is

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

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and, therefore, at a distance from C= Hence (Art. 88)

the locus is the polar of the point 0.

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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 co-ordinates, the equation becomes

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

с

ρ g cos 0+fsin '

and, returning to rectangular co-ordinates, 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 OQ 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 0 and 0', then we have pp' = k2 and 0 – 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 0; then, writing for p,

k2

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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

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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. I., p. 169).

Take the fixed point 0 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

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To find the polar co-ordinates of the point of intersection of these two, we should seek what value of 0 would render

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

Similarly, the polar co-ordinates of the intersection of the first and third circles are 0 = a + 2, and p = d cos a cosy.

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 and p, and we shall get two equations to determine p and k. We shall get

Hence

p = d cosa cosẞ cos {k − (a + ẞ)} = d cos a cos y cos {k − (a + y)}.
and pd cosa cosẞ cosy.

k = a + B+

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.

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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 S+kS' = 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,

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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 y2. 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 (a' — a) x + 2 (B' — B) y + r22 − 212 + a2 − a” + ß2 — ß”2 = 0.

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 those 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.

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107. We saw (Art. 90) that if the co-ordinates of any point 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.

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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-k'S'=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 co-ordinates of the centre of Sk2S'.

Ans.

α- — k2a' ẞ — k2ß′
k2,

1

1 - k2

that is to say, the line joining the centres of S, S' is divided externally in the ratio 1: k2.

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'Ecole Polytechnique, Cahier XVI., 1813).

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