Explanation

(b)
Let centre of circle be $C( - g,\ - f)$, then equation of circle passing through origin be
$x^{2} + y^{2} + 2gx + 2fy = 0$
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$\therefore$ Distance,$d = | - g - 3| = g + 3$
In $\mathrm{\Delta}ABC,\ \ \ \ (BC)^{2} = AC^{2} + BA^{2}$
$\Rightarrow \ \ \ g^{2} + f^{2} = (g + 3)^{2} + 2^{2}$
$\Rightarrow \ \ \ g^{2} + f^{2} = g^{2} + 6g + 9 + 4$
$\Rightarrow \ \ \ f^{2} = 6g + 13$
Hence, required locus is $y^{2} + 6x = 13$