The locus of point of intersection of tangents drawn at the ends of chord of ${\mathrm{y}}^2=4\mathrm{ax}$ which subtends a right angle at vertex is 

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

Points are $\mathrm{P}({\mathrm{at}}_1^2, 2{\mathrm{at}}_1)$

$Q\left({\mathrm{at}}_2^2, 2{\mathrm{at}}_2\right)$

point of intersection of tangents at $t_1 and t_2 is (a{t}_1{t}_2,a(t_1+t_2\left)\right)$

vertex is O(0, 0)

Slope of $\overline{\mathrm{OP}} =\frac{2{\mathrm{at}}_1}{{\mathrm{at}}_1^2}=\frac{2}{{\mathrm{t}}_1}$

Slope of $\overline{OQ}=\frac{2}{{\mathrm{t}}_2}$

$$\begin{gathered} {\mathrm{m}}_1\times {\mathrm{m}}_2=-1 \\ \frac{2}{{\mathrm{t}}_1}\times \frac{2}{{\mathrm{t}}_2}=-1 \\ 4=-{\mathrm{t}}_1{\mathrm{t}}_2 \\ \Rightarrow 4a=-a{\mathrm{t}}_1{\mathrm{t}}_2 \\ \therefore 4\mathrm{a}=-\mathrm{x}\Rightarrow \mathrm{x}+4\mathrm{a}=0 \end{gathered}$$