Let $S$ be the circle in the xy-plane defined by the equation $x^2+y^2=4$ Let $P$ be a point on the circle with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at $M$ and $N$. Then the mid-point of the line segment $MN$ must lie on the curve

 

Let $P(2\cos \theta ,2\sin \theta ),u<\theta <\frac{\pi }{2}$ be a point on

$x^2+y^2=4$ The tangent to $S$ at $P$ is $x\cos \theta + y\sin \theta = 2$. This

intersects the coordinate axes at $M\left(\frac{2}{\cos \theta },0\right)$ and 

$N\left(0,\frac{2}{\sin \theta }\right)$ Let $Q(h, k)$ be the mid-point of $MN$. Then,

$h=\frac{1}{\cos \theta },k=\frac{1}{\sin \theta }\Rightarrow \frac{1}{h^2}+\frac{1}{k^2}=1\Rightarrow h^2+k^2=h^2{k}^2$

Hence, the locus of $Q (h, k)$ is $x^2+y^2=x^2{y}^2$