The locus of the midpoint of the chords of the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=16,$ which are tangents to the hyperbola $9{\mathrm{x}}^2-16{\mathrm{y}}^2=144,$ is 
 

Given circle ${\mathrm{x}}^2+{\mathrm{y}}^2=16$. 

let $\mathrm{P}({\mathrm{x}}_1, {\mathrm{y}}_1)$ be locus of mid point of chord of  the circle.

Now equation of chord is ${\mathrm{S}}_1={\mathrm{S}}_{11}$

$\Rightarrow {\mathrm{xx}}_1+{\mathrm{yy}}_1 -({{\mathrm{x}}_1}^2+{{\mathrm{y}}_1}^2)=0 \to \left(1\right)$

Given hyperbola $\frac{{\mathrm{x}}^2}{16}-\frac{{\mathrm{y}}^2}{9}=1.$

If $\left(1\right)$ is tangent to  the hyperbola then $n^2={\mathrm{a}}^2{\mathrm{l}}^2-{\mathrm{b}}^2{\mathrm{m}}^2$ $(\because$ condition of tangent)

$$\begin{gathered} where l=x_1,m=y_1,n=-({x^2}_1+{y^2}_1) \\ \Rightarrow {({{\mathrm{x}}_1}^2+{{\mathrm{y}}_1}^2)}^2=16{{\mathrm{x}}_1}^2-9{{\mathrm{y}}_1}^2 \end{gathered}$$

$\therefore$ locus is $16{\mathrm{x}}^2-9{\mathrm{y}}^2={({\mathrm{x}}^2+{\mathrm{y}}^2)}^2$