A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis,

Let equation of circle is

$x^{2}+y^{2}+2 f x+2 f y+e=0$, it passes through $(0,2 b)$

$\Rightarrow 0+4 b^{2}+2 g \times 0+4 f+c=0$

$\Rightarrow 4 b^{2}+4 f+c=0$             ........(1)

$2 \sqrt{\mathrm{g}^{2}-\mathrm{c}}=4 \mathrm{a}$   .....(2)

$\mathrm{g}^{2}-\mathrm{c}=4 \mathrm{a}^{2} \Rightarrow \mathrm{c}=\left(\mathrm{g}^{2}-4 \mathrm{a}^{2}\right)$

Putting in equation (1)

$\Rightarrow 4 b^{2}+4 f+g^{2}-4 a^{2}=0$

$\Rightarrow x^{2}+4 y+4\left(b^{2}-a^{2}\right)=0$, it represent a parabola.