If a tangent to the circle

Let any tangent to circle $x^{2}+y^{2}=1$ is $x \cos \theta+y \sin \theta=1$

Since, $P$ and $Q$ are the point of intersection on the coordinate axes.

Then $P \equiv\left(\frac{1}{\cos \theta}, 0\right) \& Q \equiv\left(0, \frac{1}{\sin \theta}\right)$

$\therefore$ mid-point of PQ be $M \equiv\left(\frac{1}{2 \cos \theta}, \frac{1}{2 \sin \theta}\right) \equiv(h, k)$

$\Rightarrow \cos \theta=\frac{1}{2 h}$..........(1)

$\sin \theta=\frac{1}{2 k}$ .............(2)

Now squaring and adding equation (1) and (2)

$\frac{1}{h^{2}}+\frac{1}{k^{2}}=4$

$\Rightarrow \mathrm{h}^{2}+\mathrm{k}^{2}=4 \mathrm{~h}^{2} \mathrm{k}^{2}$

$\therefore$ locus of $M$ is $: x^{2}+y^{2}=4 x^{2} y^{2}$