Tangents are drawn from any point on the hyperbola $ \frac{x^2}{9}– \frac{y^2}{4} = 1$ to the circle $x^2 + y^2 = 9$. Find the locus of mid point of the chord of contact.
I tried the following. Let the locus of mid point of contact is $hx+ky=h^2+k^2$. ( General formula)
Let $h=(x_1+x_2)/2$ & $k=(y_1+y_2)/2$ Let $(x_2,y_2)$ be tangent of circle point and $(x_1,y_1)$ are point on hyperbola from which tangent is drawn to the circle. I mapped tangent as $xx_2+yy_2=9$ and putting $(x_1,y_1)$ we get $x_1x_2+y_1y_2=9$ after that I am struck.
WLOG any point on the hyperbola, $$(3\sec t,2\tan t )$$
From this, the equation of chord of contact $$x(3\sec t)+y(2\tan t)-9=0$$
Comparing with $$xh+yk-(h^2+k^2)=0$$
$$\dfrac{3\sec t}h=\dfrac{2\tan t}k=\dfrac9{h^2+k^2}$$
Again, $$\dfrac{\sec t}{\dfrac h3}=\dfrac{\tan t}{\dfrac k2}=\pm\sqrt{\dfrac{\sec^2t-\tan^2t}{\left(\dfrac h3\right)^2-\left(\dfrac k2\right)^2}}=\pm\dfrac6{\sqrt{4h^2-9k^2}}$$
Replace the value of $\sec t$ in $$\dfrac{3\sec t}h=\dfrac9{h^2+k^2}$$
Then square both sides and simplify