Explanation

(b)
Let the equation of circle is
$x^{2} + y^{2} + 2gx + 2fy + c = 0$
It cuts the circle $x^{2} + y^{2} = 4$ orthogonally, if
$2g.0 + 2f.0 = c - 4\ \ \Rightarrow \ \ \ c = 4$
$\therefore$ equation of circle is
$x^{2} + y^{2} + 2gx + 2fy + 4 = 0$
$\because$ It passes through the points $(a,\ b)$
$\therefore\ \ \ a^{2} + b^{2} + 2ag + 2fb + 4 = 0$
Locus of centre $( - g,\ - f)$ will be
$a^{2} + b^{2} - 2xa - 2yb + 4 = 0$
$\Rightarrow \ 2ax + 2by - \left( a^{2} + b^{2} + 4 \right) = 0$