Explanation

(c)
The radial axis of the Two given circles is
$2x\left( g - \frac{3}{4} \right) + 2y(f - 2) = 0\ \ \ \ \lbrack\because S_{1} - S_{2} = 0\rbrack$
$\Rightarrow \ \ x\left( g - \frac{3}{4} \right) + y(f - 2) = 0$
It touches the circle
$x^{2} + y^{2} + 2x + 2y + 1 = 0$
$\ \therefore\ \ \left| \frac{- \left( g - \frac{3}{4} \right) - (f - 2)}{\sqrt{\left( g - \frac{3}{4} \right)^{2} + (f - 2)^{2}}} \right| = 1$
$\Rightarrow \ \ \left( g - \frac{3}{4} \right)^{2} + (f - 2)^{2} + 2\left( g - \frac{3}{4} \right)(f - 2) = \left( g - \frac{3}{4} \right)^{2} + (f - 2)^{2}$
$\Rightarrow \ \ \left( g - \frac{3}{4} \right)(f - 2) = 0$
$\Rightarrow \ \ g = \frac{3}{4}or\ \ f = 2$