Find the equation to the circle cutting orthogonally the three circles $x^{2} + y^{2} = a^{2}$ , ${(x - c)}^{2} + y^{2} = a^{2}$ , and $x^{2} + {(y - b)}^{2} = a^{2}$ .

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Solution

Let the circle be $x^{2} + y^{2} + 2 g x + 2 f y + d = 0$
For two circles to be orthogonal, $2 g g^{'} + 2 f f^{'} = c + c^{'}$
So from the question,
$d - a^{2} = 0....... (1)$
$2. g . (- c) = d + c^{2} - a^{2} . . . . . . . . (2)$
$2. f . (- b) = d + b^{2} - a^{2} . . . . . . . . . . (3)$
Solving $(1), (2)$ and $(3)$ , we get
$g = - \frac{c}{2}$
$f = - \frac{b}{2}$
$d = a^{2}$
So the circle is,
$x^{2} + y^{2} - c x - b y + a^{2} = 0$

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