The equation of the director circle to the parabola $y^2=4ax$ is 

The equations of tangents at P and Q are $y{t}_1=x+a{t}_1^2$ and $y{t}_2=x+a{t}_2$

The point of intersection of these tangents is 

$\left[a{t}_1{t}_2,a\left(t_1+t_2\right)\right]$

Let this point be $(h,k)$

The slope of the tangents are $m_1=\frac{1}{t_1}\text{ and }{m}_2=\frac{1}{t_2}$

Since these two tangents are perpendicular, so

$\begin{array}{l}{m}_1{m}_2=-1\\ \Rightarrow \frac{1}{t_1}\cdot \frac{1}{t_2}=-1\\ \Rightarrow t_1\cdot t_2=-1\\ \Rightarrow \frac{h}{a}=-1\\ \Rightarrow h=-a\end{array}$

Thus the locus of the points of intersection is $x+a=0$

which is the directrix of the parabola $y^2=4ax$