A circle cuts a chord of length 4 a on the

Let centre be $C(h, k)$

$C Q=C P=r$

$\Rightarrow \quad C Q^{2}=C P^{2}$

$(h-0)^{2}+(k \pm 0)^{2}=C M^{2}+M P^{2}$

$h^{2}+(k \pm 2 b)^{2}=k^{2}+4 a^{2}$

$h^{2}+k^{2}+4 b^{2} \pm 4 b k=k^{2}+4 a^{2}$

Then, the locus of centre $C(h, k)$

$x^{2}+4 b^{2} \pm 4 b y=4 a^{2}$

Hence, the above locus of the centre of circle is a parabola.