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A circle cuts a chord of length 4 a on the $x$ -axis and passes through a point on the $y$ -axis, distant $2 \mathrm{~b}$ from the origin. Then the locus of the centre of this circle, is:
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The correct answer is:
a parabola
Let centre be $C(h, k)$
$C Q=C P=r$
$\Rightarrow C Q^{2}=C P^{2}$
$(h-0)^{2}+(k \pm 0)^{2}=C M^{2}+M P^{2}$
$h^{2}+(k \pm 2 \mathrm{~b})^{2}=k^{2}+4 a^{2}$
$h^{2}+k^{2}+4 b^{2} \pm 4 b k=k^{2}+4 a^{2}$
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.
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