Explanation

(c)
Given, $\frac{|x + (y + 1)i|}{|x + (y - 1)i|} = \sqrt{3}$
$\Rightarrow \ \ \ \ x^{2} + (y + 1)^{2} = 3\lbrack x^{2} + (y - 1)^{2}\rbrack$
$\Rightarrow \ \ \ \ x^{2} + y^{2} - 4y + 1 = 0$
On comparing with
$x^{2} + y^{2} + 2gx + 2fy + c = 0$, we get
$g = 0,\ \ \ f = - 2,\ \ \ c = 1$
$\therefore$ Radius of circle$= \sqrt{g^{2} + f^{2} - c} = \sqrt{( - 2)^{2} - 1} = \sqrt{3}$