Let the equation of the circle be
$$
x^2+y^2+2 g x+2 f y+c=0
$$
It passes through origin
$$
\text { so } c=0
$$
Then, the equation of circle is
$$
x^2+y^2+2 g x+2 f y=0
$$
It also passes through $A\left(x_1, 0\right)$
$$
\begin{array}{rlrl}
\therefore & x_1^2+0+2 g x_1 & =0 \\
& \Rightarrow & g & =-\frac{x_1}{2}
\end{array}
$$
It also passes through $B\left(0, y_1\right)$
$$
\therefore o+y_1^2+2 f y_1=0 \Rightarrow f=\frac{-y_1}{2}
$$
$\Rightarrow$ centre of the circle is $\left(\frac{x_1}{2}, \frac{y_1}{2}\right)$
Mid point of $A B$ is $\left(\frac{x_1}{2}, \frac{y_1}{2}\right)$
$$
\text { i.e., } O M=A M=B M
$$
Thus,
$$
\begin{aligned}
c=O M \Rightarrow c & =\sqrt{\frac{x_1^2}{2}+\frac{y_1^2}{2}} \\
x_1^2+y_1^2 & =4 c^2
\end{aligned}
$$
Thus, locus is $x^2+y^2=4 c^2$