Explanation

(c)
The equation of the common chord of the circles $(x - a)^{2} + y^{2} = a^{2}$ and $x^{2} + (y + b)^{2} = b^{2}$ is
$I \equiv S_{1} - S_{2} = 0$
$\Rightarrow x^{2} + a^{2} - 2ax + y^{2} - a^{2} - x^{2} - y^{2} - b^{2} - 2by + b^{2} = 0$
$\Rightarrow ax + by = 0$ …(i)
Now, the equation of required circle is
$S_{1} + \lambda L = 0$
$\therefore\left\{ (x - a)^{2} + y^{2} - a^{2} \right\} + \lambda\left\{ ax + by \right\} = 0$
$\Rightarrow x^{2} + y^{2} + x(a\lambda - 2a) + \lambda by = 0$ …(ii)
Since, Eq. (i) is a diameter of Eq. (ii), then
$a\left( - \frac{a\lambda - 2a}{2} \right) + b\left( - \frac{\lambda b}{2} \right) = 0$
$\Rightarrow \ \lambda = \frac{2a^{2}}{a^{2} + b^{2}}$
On putting the value of $\lambda$ in Eq. (ii), we get
$\left( a^{2} + b^{2} \right)\left( x^{2} + y^{2} \right) = 2ab(bx - ay)$
Which is the required equation of circle