Tangent is drawn at any point (p, q) on the parabola y² = 4ax . Tangents are drawn from any point on this tangent to the circle x² + y² = a² , such that the chords of contact passes through a fixed point (r, s). Then p, q, r and s can hold the relation

The equation of tangent is $\mathrm{yq}=2\mathrm{a}(\mathrm{x}+\mathrm{p})\text{. Let }\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)\text{ lies on this tangent. }$
Then ${\mathrm{y}}_1\mathrm{q}=2\mathrm{a}\left({\mathrm{x}}_1+\mathrm{p}\right)$ Chord of contact for the circle ${\mathrm{xx}}_1+{\mathrm{yy}}_1={\mathrm{a}}^2$ 
 Passing through (r, s) 
$$\begin{gathered} {\mathrm{rx}}_1+{\mathrm{sy}}_1={\mathrm{a}}^2 \\ \mathrm{Also},{\mathrm{xx}}_1+\mathrm{y}\frac{2\mathrm{a}}{\mathrm{q}}\left({\mathrm{x}}_1+\mathrm{p}\right)-{\mathrm{a}}^2=0 \\ {\mathrm{x}}_1\left(\mathrm{x}+\frac{2\mathrm{ay}}{\mathrm{q}}\right)+\left(\frac{2\mathrm{ap}}{\mathrm{q}}\mathrm{y}-{\mathrm{a}}^2\right)=0 \end{gathered}$$
$$\begin{gathered} {\mathrm{x}}_1(\mathrm{qx}+2\mathrm{ay})+\left(2\mathrm{apy}-{\mathrm{a}}^2\mathrm{q}\right)=0 \\ \Rightarrow \mathrm{qx}+2\mathrm{ay}=0\text{ and }2\mathrm{apy}={\mathrm{a}}^2\mathrm{q} \end{gathered}$$
$$\begin{gathered} \text{ or }\mathrm{qpx}+{\mathrm{a}}^2\mathrm{q}=0\text{ or }\mathrm{x}=-\frac{{\mathrm{a}}^2}{\mathrm{p}},\mathrm{y}=\frac{\mathrm{aq}}{2\mathrm{p}} \\ \text{ So, }\mathrm{r}=-\frac{{\mathrm{a}}^2}{\mathrm{p}}\text{ and }\mathrm{s}=\frac{\mathrm{aq}}{2\mathrm{p}} \\ \text{ ⇒ }4{\mathrm{ps}}^2=-{\mathrm{rq}}^2 \end{gathered}$$