SOLUTION —
Let $(h, k)$ be the point.
$\begin{aligned}\because & 4 \sqrt{(h-h)^2+k^2} & =h^2+k^2 \\\Rightarrow & 4|k| & =h^2+k^2\end{aligned}$
Locus of the point is
$\begin{aligned}4|y| & =x^2+y^2 \\\Rightarrow \quad x^2+y^2-4|y| & =0\end{aligned}$
So, The correct option will be (B).