If the chord of contact of tangents from a point $P\left(x_1,y_1\right)$ to the circle $x^2+y^2=a^2$ touches the circle $(x-a{)}^2+y^2=a^2$, then the locus of 

$\left(x_1,y_1\right)$ is

The equation of the chord of contact of tangents

drawn from a point $P\left(x_1,y_1\right)$ to $x^2+y^2=a^2$ is $x{x}_1+y{y}_1=a^2.$

It will touch $(x-a{)}^2+y^2=a^2,$if

$\begin{array}{l}\left|\frac{a{x}_1+0y_1-a^2}{\sqrt{x_1^2+y_1^2}}\right|=a\\ \Rightarrow x_1-a=\pm \sqrt{x_1^2+y_1^2}\\ \Rightarrow {\left(x_1-a\right)}^2=x_1^2+y_1^2\Rightarrow y_1^2=a^2-2a{x}_1\end{array}$

Hence, the locus of $\left(x_1,y_1\right)$is $y^2=a^2-2ax$ which is a parabola.