Let the two tangents to the parabola \(y^{2}=4 a x\) be \(P T\) and \(Q T\) which are at right angle to one another at \(\mathrm{T}(\mathrm{h}, \mathrm{k})\).

Then we have a find the locus of \(\mathrm{T}(\mathrm{h}, \mathrm{k})\).

We know that \(y=m x+\frac{a}{m}\), where \(m\) is the slope is the equation of tangent to the parabola \(\mathrm{y}^{2}=4 \mathrm{ax}\) for all \(\mathrm{m}\).

Since this tangent to the parabola will pass through \(\mathrm{T}(\mathrm{h}, \mathrm{k})\), so

\(\mathrm{k}=\mathrm{mh}+\frac{\mathrm{a}}{\mathrm{m}} ; \text { or } \mathrm{m}^{2} \mathrm{~h}-\mathrm{mk}+\mathrm{a}=0\)

This is a quadratic equation in \(m\), so will have two roots, say \(m_{1}\) and \(m_{2}\), then \(\mathrm{m}_{1}+\mathrm{m}_{2}=\frac{\mathrm{k}}{\mathrm{h}}\), and \(\mathrm{m}_{1} \cdot \mathrm{m}_{2}=\frac{\mathrm{a}}{\mathrm{h}}\)

Given that the two tangents intersect at right angle so

\(\mathrm{m}_{1} \cdot \mathrm{m}_{2}=-1 \text { or } \frac{\mathrm{a}}{\mathrm{h}}=-1 \text { or } \mathrm{h}+\mathrm{a}=0\)

The locus of \(\mathrm{T}(\mathrm{h}, \mathrm{k})\) is \(\mathrm{x}+\mathrm{a}=0\), which is the equation of directrix.