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tangents. If the origin be the centre of the circle (S), then a and B=0; and we find, for the equations of the chords of contact,

a'x+ B'y = r (r Fr').

Ex. 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

2x+y=6, 2x + y = 3.

The first chord meets the circle in the points (2, 2), (, ), the tangents at which are y=2, 4x-3y = 10,

and the second chord meets the circle in the points (1, 1), (3, 3), the tangents at which are

x=1, 8x+4y = 5.

114. 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 coordinates are easily found, for O is the pole, with regard to circle (S), of the chord AA', whose equation is

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Comparing this equation with the equation of the polar of the point x'y',

(x' — α) (x − a) + (y' -- B) (y — B) = r2,

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(a'-a) r

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(B' - B) r

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So, likewise, the coordinates of O' are found to be

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These values of the coordinates 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

x2+ y2-6x-8y = 0, x2 + y2 - 4x-6y=8.

The equation of the pair of tangents through x'y' to

is found (Art. 92) to be

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{(x' — a)2 + (y' — f3)2 — r2} {(x − a)2 + (y − (?)2 — p2} = {(x − a) (x' — a) + (y — B) (y' — B) — r??

P

Now the coordinates 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 two common tangents become imaginary; but their equation is found, by calculating the pair of tangents through the other centre of similitude (, ), to be

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

115. 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 me, 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. 84, Ex. 2)

x2 + y2+ 2xy cosa - 2ax - 2a y + a2 = 0,

12

x2+ y2+ 2xy cosw-2a'x - 2a'y + a22 = 0.

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

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

ах ay
a' a

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.

=

116. 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', σ, 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.

Take OR, OS for axes, then we saw (Art. 115) that OR=mOp, OS=mOo, and that if the equation of the circle pop'o' be

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

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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

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2 (gx+fy) + (m + 1) c = 0,

the radical axis of the two circles.

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

117. Given three circles S, S', S"; the line joining a centre of similitude of S and S' to a centre of similitude of S and S" will pass through a centre of similitude of S' and S".

Form the equation of the line joining the first two of the points (ra-ar' rß'-Br" 'ra"-ar" rẞ"-Br" 'r'a"-r"a' r'ß"—r"B"

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(Art. 114), and we get (see Ex. 6, p. 24),

{r (B′ – B") + r' (B′′ – B) + r" (B — B')} ∞

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=r (B'a” – B'a') + 2o′ (B"a - Ba") + r" (Ba' – B'a).

Now the symmetry of this equation sufficiently shows, that the
line it represents must pass through the third centre of similitude.
This line is called an axis of similitude of the three circles.
Since for each pair of

circles there are two centres of similitude, there will be in all six for the three circles, and these will be distributed along four axes of similitude, as represented in the figure. The equations of the other three will be found by changing the signs of either r, or r', or r", in the equation just given.

COR. If a circle (E) touch two others (S and S'), the line joining the points of contact will pass through a centre of similitude of S and S'. For when two circles touch, one of their centres of similitude will coincide with the point of contact.

If Σ touch S and S', either both externally or both internally, the line joining the points of contact will pass through the external centre of similitude of S and S'. If Σ touch one externally and the other internally, the line joining the points of contact will pass through the internal centre of similitude.

*118. To find the locus of the centre of a circle cutting three given circles at equal angles.

If a circle whose radius is R, cut at an angle a the three circles S, S', S", then (Art. 112, Ex. 8) the coordinates of its centre fulfil the three conditions

S= R* — 2 Rr cosa, S'= R2 - 2Rr' cosa, S" = R2 - 2 Rr" cosa.

From these conditions we can at once eliminate Rand R cosa. Thus, by subtraction,

whence

S-S' = 2R (r' − r) cosa, S-S" = 2R (r" − r) cosa,

( S − S'') (r — r') = ( S − S′′) (r — r'),

the equation of a line on which the centre must lie. It obviously

passes through the radical centre (Art. 108); and if we write for S-S', S-S", their values (Art. 105), the coefficient of x in the equation is found to be

-2 {a (r'-') + a' (r" − r) + a" (r — r')},

while that of y is

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Now if we compare these values with the coefficients in the equation of the axis of similitude (Art. 117), we infer (Art. 32), that the locus is a perpendicular let fall from the radical centre on an axis of similitude.

It is of course optional which of two supplemental angles we consider to be the angle at which two circles intersect. The formula (Art. 112) which we have used assumes that the angle at which two circles cut is measured by the angle which the distance between their centres subtends at the point of meeting; and with this convention, the locus under consideration is a perpendicular on the external axis of similitude. If this limitation be removed, the formula we have used becomes S-R*+2Rr cosa; or, in other words, we may change the sign of either r, r', or " in the preceding formulæ, and therefore (Art. 117) the locus is a perpendicular on any of the four axes of similitude.*

When two circles touch internally, their angle of intersection vanishes, since the radii to the point of meeting coincide. But if they touch externally, their angle of intersection according to the preceding convention is 180°, one radius to the point of meeting being a continuation of the other. It follows, from

In fact, all circles cutting three circles at equal angles have one of the axes of similitude for a common radical axis. Let E, E', E" be three circles, all cutting the given circles at the same angles a, ß, y respectively. Then the coordinates of the centre of each of the circles S, S', S" must fulfil the conditions

Σ=r22rR cosa, Σ'22rR' cos ß, Σ"= r2 - 2rR" cos y ;

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whence (R cos a - R" cos y) (Σ — Σ') = (R c03 a - R' cos ẞ) (Σ – Σ"). This which appears to be the equation of a right line is satisfied by the coordinates of the centre of S, of S', and of S", three points which are not supposed to be on a right line. Now the only way in which what seems an equation of the first degree, such as ax + by + c = a'x + b'y + c' can be satisfied by the coordinates of three points which are not on a right line, is if the equation is in truth an identical one, a = a', bb', cc. The equation, therefore, written above denotes an identical relation of the form Σ=kΣ'+E", shewing that the three circles have a common radical axis.

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