The locus of the poles with respect to the parabola $y^2 = 4ax$ of the tangent to the hyperbola $x^2 - y^2=a^2$is 

Let $(h, k)$ be the pole. The equation of the polar from the point $(h, k)$ w.r.t. the parabola $y^2 = 4ax$ is 

  $yk=2a(x+h)=2ax+2ah$

$\Rightarrow y=\left(\frac{2a}{k}\right)x+\left(\frac{2ah}{k}\right) ...\left(i\right)$

If the line $\left(i\right)$ be a tangent to the hyperbola $x^2 - y^2 = a^2$ then

$\begin{array}{l} c^2=a^2{m}^2-a^2\\ \Rightarrow {\left(\frac{2ah}{k}\right)}^2=a^2{\left(\frac{2a}{k}\right)}^2-a^2\\ \Rightarrow 4h^2=4a^2-k^2\\ \Rightarrow h^2+\frac{k^2}{4}=a^2\end{array}$

Hence, the locus of $(h, k)$ is 

   $\begin{array}{l} x^2+\frac{y^2}{4}=a^2\\ \Rightarrow 4x^2+y^2=4a^2\end{array}$