Tangents are drawn from a point $P$ to the parabola $y^2 = 4ax$. If the chord of contact of the parabola be a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, find the locus of the point $P$.

Let the point on the parabola be $(h, k)$.

The equation of the chord of contact of the parabola is 

  $yk=2a(x+h)$      …(i)

$\Rightarrow y=\frac{2a}{k}x+\frac{2ah}{k}$    …(ii)

Since, the line (ii) is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, so, 

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

Hence, the locus of $(h, k)$ is 

$4a^2{x}^2=4a^4-y^2{b}^2$