Explanation

(c)
Let the equation of the circle be
$x^{2} + y^{2} + 2gx + 2fy + c = 0$
Since, this passes through (1, 2)
$\therefore\ \ 1^{2} + 2^{2} + 2g(1) + 2f(2) + c = 0$
$\Rightarrow 5 + 2g + 4f + c = 0$ …(i)
Also, the circle $x^{2} + y^{2} = 4$ intersects the circle
$x^{2} + y^{2} + 2gx + 2fy + c = 0$ orthologonally
$\therefore\ \ \ 2(g.0 + f.0) = c - 4\ \ \Rightarrow \ \ \ c = 4$
On putting the value of $c$ in Eq. (i), we get
$2g + 4f + 9 = 0$
Hence, the locus of centre $( - g,\ - f)$ is
$- 2x - 4y + 9 = 0\ \ \ \Rightarrow \ \ 2x + 4y - 9 = 0$