Ex. 4.. If a moveable circle cut two fixed circles at constant angles, it will cut all circles having the same radical axis at constant angles.

R2 2Rr cos a =

Let the equations of the two fixed circles be S = 0, S'= 0, and their radii r,r'; then the co-ordinates of the centre of the moveable circle fulfil the relations, S, R22Rr' cosẞ = S', since D2 - r2 = the square of the tangent to the first fixed circle = we have kS+ IS' k + l

R22R

kr cos alr' cosß
k + l

S (Art. 88). Then,

which is precisely the condition that the moveable circle should cut the circle kS+ IS' at the constant angle y; where (k + 1)r" cosy = kr cos a + lr' cos ß, r" being the radius of the circle kS+ IS'

Ex. 5. A circle which cuts two fixed circles at constant angles will also touch two fixed circles.

For we can determine the ratio k: 1, so that y shall = 0, or cos y = 1. It will easily be found that if D be the distance between the centres of S and S',

(k + 1)2 r′′2 = (k + 1) (kr2 + lr′2) — klD2.

Substituting this value for r" in the equation of the last example, we get a quadratic to determine k: l.

118. To draw a common tangent to two circles. Let their equations be

and

(x − a )2 + (y − b )2 = p2

(S),

(xa')2+(y- b')" = p12

(S').

We saw (Art. 83) that the equation of a tangent to (S) was

(x − a) (x' − a) + (y − b) (y' − b) = r2 ;

In like manner, any tangent to (S) is

(x - a) cosẞ + (y - b) sin ẞ= r'.

tan B,

Now, if we seek the conditions necessary that these two equations should represent the same right line; first, from comparing the ratio of the coefficients of x and y, we get tan a = whence ẞ either = a, or= 180° + a. If either of these conditions be fulfilled, we must equate the absolute terms, and we find, in the first case,

(a - a') cos a + (b − b') sin a + r − r′ = 0,

and in the second case,

(a− a) cos a + (b − b') sin a + r + r' = 0.

Either of these equations would give us a quadratic to deterThe two roots of the first equation would correspond

mine a.

to the direct or exterior common tangents, Aa, A'a'; the roots of the second equation would correspond to the transverse or interior tangents, Bb, B'b'.

If we wished to find the co-ordinates of the point of contact of the common tangent with the circle (S), we must substitute,

in the equation just found, for cos a, its value,

or else,

and we find

α

and for

(a − a) (x' − a) + (b − b') (y' − b) + r (r − r') = 0 ;

(a − a') (x' − a) + (b − b') (y' − b) + r (r + r') = 0.

The first of these equations, combined with the equation (S) of the circle, will give a quadratic, whose roots will be the coordinates of the points A and A', in which the direct common tangents touch the circle (S); and it will appear, as in Art. 86, that

(a' - a) (x − a) + (b' − b) (y − b) = r (r − r')

is the equation of AA', the chord of contact of direct common tangents. So, likewise,

(a' − a) (x − a) + (b' − b) (y − b) = r (r + r)

is the equation of the chord of contact of transverse common tangents. If the origin be the centre of the circle (S), then a and b = 0; and we find, for the equation of the chord of contact,

a' x + b'y = r (r = r').

Ex. 1. Find the common tangents to the circles

x2 + y2 — 4x — 2y + 4 = 0,

x2 + y2 + 4x + 2y - 4 = 0.

The chords of contact of common tangents with the first circle are

119. The points O and O', in which the direct or transverse tangents intersect, are (for a reason explained in the next Article) called the centres of similitude of the two circles.

Their co-ordinates are easily found, for O is the pole, with regard to circle (S), of the chord AA', whose equation is

Comparing this equation with the equation of the polar of the

So, likewise, the co-ordinates of O' are found to be

These values of the co-ordinates indicate (see Art. 7) that the centres of similitude are the points where the line joining the centres is cut externally and internally in the ratio of the radii.

Ex. Find the common tangents to the circles

{(x-a)2+(y'-b)—ra} {(x− a)2+(y—b)2—r2} = {(x− a) (x'-a)+(y- b) (y'—b) – r2}1.

Now, the co-ordinates of the exterior centre of similitude are found to be (-2, -1), and hence the pair of tangents through it is

25 (x2+ y2 — 6x — 8y) = (5x + 5y — 10)2; or xy + x + 2y + 2 =0; or (x + 2) (y + 1)= 0. As the given circles intersect in real points, the other pair of common tangents become imaginary; but their equation is found, by calculating the pair of tangents through the other centre of similitude to be

(22, 31),

9

40x2 + xy + 40y2 - 199x-278y + 722 = : 0.

120. Every right line drawn through the intersection of common tangents is cut similarly by the two circles.

It is evident that if on the radius vector to any point P there be taken a point Q, such that OP = m times OQ, then the x and y of the point P will be respectively m times the x and y of the point Q; and that, therefore, if P describe any curve, the locus of Q is found by substituting mx, my for x and y in the equation of the curve described by P.

Now, if the common tangents be taken for axes, and if we denote Oa by a, OA by a', the equations of the two circles are (Art. 82, Ex. 4)

But the second equation is what we should have found if we

ах ay
a' a'

had substituted, for x, y, in the first equation; and it

therefore represents the locus formed by producing each radius vector to the first circle in the ratio a: a'.

COR. Since the rectangle Op Op' is constant (see fig. next page), and since we have proved OR to be in a constant ratio to Op, it follows that the rectangle OR. Op' OR'. Op is constant, however the line be drawn through O.

=

121. If through a centre of similitude we draw any two lines meeting the first circle in the points R, R', S, S', and the second in the points p, p', o, o', then the chords RS, po; R'S', p'o'; will be parallel, and the chords RS, p'o'; R'S', po; will meet on the radical axis of the two circles.

Dx + Ey + (m + 1) F = 0.

Now let the equations of po and of p'o' be

S'

then the equations of RS and of R'S' must be

It is evident, from the form of the equations, that RS is parallel to po; and RS and p'o' must intersect on the line

Dx + Ey + (1 + m) F = 0,

the radical axis of the two circles.

A particular case of this theorem is, that the tangents at R ρ are parallel, and that those at R and p' meet on the radical axis.

and

122. Given three circles; the line joining a centre of similitude of the first and second to a centre of similitude of the first and third will pass through a centre of similitude of the second and third.

Form the equation of the line joining the points