The locus of the middle points of the chords of the  circle $x^2+y^2=4$ which are tangents to the hyperbola $\frac{x^2}{4}-\frac{y^2}{16}=1$. is the curve ${\left(x^2+y^2\right)}^2=\lambda x^2-16y^2$ , then the value of $\lambda$ is……

Let $p\left(h,k\right)$ be the middle point, then the equation of chord whose mid-point $p\left(h,k\right)$ is

$T=S_1$

$\Rightarrow hx+ky-4=h^2+k^2-4$

$\Rightarrow y=\frac{h}{k}x+\frac{h^2+k^2}{k}$

It will touch the hyperbola $\frac{x^2}{4}-\frac{y^2}{16}=1$, then

${\left(\frac{h^2+y^2}{k}\right)}^2=4\times \frac{h^2}{k^2}-16$

$\Rightarrow {\left(h^2+k^2\right)}^2=4h^2-16k^2$

$\because$ Locus of  $p\left(h,k\right)$is

${\left(x^2+y^2\right)}^2=4x^2-16y^2$

$\therefore \lambda =4$