Points A,B,C lie on the parabola y² = 4ax . The tangents to the parabola at A,B,C taken in pairs intersect at points P,Q,R, then, the ratio of the areas of the $∆$ABC and $∆$PQR is

Let $\left({\mathrm{at}}_1^{2}2{\mathrm{at}}_1\right),\left({\mathrm{at}}_2^{2}2{\mathrm{at}}_2\right) \mathrm{and} \left({\mathrm{at}}_3^{2}2{\mathrm{at}}_3\right)$ be the points A, B and C respectively.
$\begin{array}{l}\therefore \text{ Area of }\mathrm{△}\mathrm{ABC}=\frac{1}{2}\left|\right|\begin{array}{lll}{\mathrm{at}}_1^2& 2{\mathrm{at}}_1& 1\\ {\mathrm{at}}_2^2& 2{\mathrm{at}}_2& 1\\ {\mathrm{at}}_3^2& 2{\mathrm{at}}_3& 1\end{array}\left|={\mathrm{a}}^2\right|\left|\begin{array}{ccc}1& {\mathrm{t}}_1& {\mathrm{t}}_1^2\\ 1& {\mathrm{t}}_2& {\mathrm{t}}_2^2\\ 1& {\mathrm{t}}_3& {\mathrm{t}}_3^2\end{array}\right|\parallel \\ ={\mathrm{a}}^2\left|\left({\mathrm{t}}_1-{\mathrm{t}}_2\right)\left({\mathrm{t}}_2-{\mathrm{t}}_3\right)\left({\mathrm{t}}_3-{\mathrm{t}}_1\right)\right|\end{array}$
Coordinates of P,Q and R are
$\left({\mathrm{at}}_2{\mathrm{t}}_3,\mathrm{a}\left({\mathrm{t}}_2+{\mathrm{t}}_3\right)\right),\left({\mathrm{at}}_3{\mathrm{t}}_1,\mathrm{a}\left({\mathrm{t}}_3+{\mathrm{t}}_1\right)\right),\left({\mathrm{at}}_1{\mathrm{t}}_2,\mathrm{a}\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)\right)$ respectively, then
$\begin{array}{l}\therefore \text{ Area of }\mathrm{△}\mathrm{PQR}==\left|\begin{array}{lll}{\mathrm{at}}_2{\mathrm{t}}_3& \mathrm{a}\left({\mathrm{t}}_2+{\mathrm{t}}_3\right)& 1\\ {\mathrm{at}}_3{\mathrm{t}}_1& \mathrm{a}\left({\mathrm{t}}_3+{\mathrm{t}}_1\right)& 1\\ {\mathrm{at}}_1{\mathrm{t}}_2& \mathrm{a}\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)& 1\end{array}\right|=\frac{{\mathrm{a}}^2}{2}\left|\begin{array}{lll}{\mathrm{t}}_2{\mathrm{t}}_3& {\mathrm{t}}_2+{\mathrm{t}}_3& 1\\ {\mathrm{t}}_3{\mathrm{t}}_1& {\mathrm{t}}_3+{\mathrm{t}}_1& 1\\ {\mathrm{t}}_1{\mathrm{t}}_2& {\mathrm{t}}_1+{\mathrm{t}}_2& 1\end{array}\right|\\ =\frac{1}{2}\times \text{ Area of }\mathrm{△}\mathrm{ABC}\\ \frac{\text{ Area of }\mathrm{△}\mathrm{ABC}}{\text{ Area of }\mathrm{△}\mathrm{PQR}}=2\end{array}$