The line  $x\cos \alpha +y\sin \alpha =p$ is a tangent to $y^2=4ax$ if 

$x\cos \alpha +y\sin \alpha =p\Rightarrow y=(-\cot \alpha )x+p\mathrm{cosec}\alpha$

The lines $y=mx+\frac{a}{m}$is a tangent 

$\therefore m=-\cot \alpha ,\frac{a}{m}=p\mathrm{cosec}\alpha$

Eliminating, $m,a=-p\cot \alpha \mathrm{cosec}\alpha =\frac{-p\cos \alpha }{{\sin }^2\alpha }$

$\therefore p\cos \alpha +a{\sin }^2\alpha =0$