$S_1=S_{11}$A variable chord of the circle $x^2 + y^2 = a^2$ touches the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.The locus of its mid-points is.

If $(h, k)$ be the mid-point of the chord of the circle $x^2+y^2=a^2$, then

  $S_1=S_{11}$

$\begin{array}{l}\Rightarrow hx+ky=h^2+k^2\end{array}$

$\Rightarrow y=\left(-\frac{h}{k}\right)x+\left(\frac{h^2+k^2}{k}\right)$     (i)

If the line (i) be a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, so, $c^2=a^2{m}^2-b^2$

$\begin{array}{l}\Rightarrow {\left(\frac{h^2+k^2}{k}\right)}^2=a^2\left(\frac{h^2}{k^2}\right)-b^2\\ \Rightarrow {\left(h^2+k^2\right)}^2=\left(a^2{h}^2-b^2{k}^2\right)\end{array}$

Hence, the locus of $(h, k)$ is

${\left(x^2+y^2\right)}^2=\left(a^2{x}^2-b^2{y}^2\right)$.