Show that the locus of the point of intersection of perpendicular tangents to the parabola y² = 4ax is the directrix x + a = 0.

Any tangent to y² = 4ax in slope form is $\mathrm{y}=\mathrm{mx}+\frac{\mathrm{a}}{\mathrm{m}}\Rightarrow {\mathrm{m}}^2\mathrm{x}-\mathrm{my}+\mathrm{a}=0$
If tangents are drawn from external point $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)\text{ then, }{\mathrm{m}}^2{\mathrm{x}}_1-{\mathrm{my}}_1+\mathrm{a}=0$
Let roots are ${\mathrm{m}}_1\text{ and }{\mathrm{m}}_2\text{ then }{\mathrm{m}}_1{\mathrm{m}}_2=\mathrm{a}/{\mathrm{x}}_1$
Since given tangents are perpendicular
$\begin{array}{l}{\mathrm{m}}_1{\mathrm{m}}_2=-1\\ \Rightarrow \frac{\mathrm{a}}{{\mathrm{x}}_1}=-1\Rightarrow {\mathrm{x}}_1=-\mathrm{a}\Rightarrow {\mathrm{x}}_1+\mathrm{a}=0\end{array}$
Locus of (x₁, y₁) is x + a = 0 which is directrix of parabola.