Let the equation of circle

$x^2+y^2=r^2$

Let $M\left(r\cos \theta ,r\sin \theta \right)$

Equation of tangents at

$M\left(r\cos \theta ,r\sin \theta \right)$ is $rx\cos \theta +ry\sin \theta =r^2$

$\Rightarrow x\cos \theta +y\sin \theta =r$

Tangent is passes through $P$.

$\therefore x\cos \theta +y\sin \theta =r$

$\Rightarrow y=\frac{r\left(1-\cos \theta \right)}{\sin \theta }=r\tan \frac{\theta }{2}$

$PQ$ is parallel to $AB.$

$\therefore k=r\tan \frac{\theta }{2}$

$OQ$ is normal of circle.

$\therefore$ Slope of normal $=\frac{k}{h}=\tan \theta$

$\Rightarrow \frac{k}{h}=\frac{2\tan \frac{\theta }{2}}{1-{\tan }^2\frac{\theta }{2}}$

$\Rightarrow \frac{k}{h}=\frac{2\frac{k}{r}}{1-\frac{k^2}{r^2}} \left[\because \tan \frac{\theta }{2}=\frac{k}{r}\right]$

$\Rightarrow 2hr=r^2-k^2\Rightarrow k^2=r^{2}2hr$

Locus of $Q$ is $y^2=-2r\left(x-\frac{r}{2}\right)$

which is parabola.