The locus of point of intersection of perpendicular tangents drawn to ${\mathrm{y}}^2=4\mathrm{ax}$ is

${\mathrm{y}}^2=4\mathrm{ax}$

The slope form of tangent to the parabola is $y=mx+\frac{a}{m}$

This can be writte as

$my-m^{2}x-a=0\Rightarrow m^{2}x-my+a=0$

If this is passing  through the point $\left(x_1,y_1\right)$ then 

$m^2{x}_1-m{y}_1+a=0$

The product of the slopes is $m_1{m}_2=\frac{a}{x_1}$

The locus of point of intersection of tangents which are pependicular is $m_1{m}_2=-1$

$\frac{a}{x_1}=-a\Rightarrow x_1+a=0$

Perpendicular tangents intersect on a directrix whose equations is $\boxed{\mathbf{x}\mathbf{+}\mathbf{a}\mathbf{=}\mathbf{0}}$