The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix x = -3 are in the ratio 3:1. If R$(\mathrm{\alpha },\mathrm{\beta })$ is the point of intersection of the tangents to the parabola at P and Q then  $\frac{{\mathrm{\beta }}^2}{\mathrm{\alpha }}$ is equal to ……

S = (3, 0) $\mathrm{p}=\left({\mathrm{at}}_1^2, 2{\mathrm{at}}_1\right)$
Directrix x = -3 $\mathrm{Q}=\left({\mathrm{at}}_2^2, 2{\mathrm{at}}_2\right)$
$\frac{2{\mathrm{at}}_1}{2{\mathrm{at}}_2}=\frac{3}{1}\Rightarrow \frac{{\mathrm{t}}_1}{{\mathrm{t}}_2}=\frac{3}{1}\Rightarrow t_1 = 3t_2$
$\begin{array}{l}{\mathrm{y}}^2=4\mathrm{ax}\\ \mathrm{a}=3\Rightarrow {\mathrm{y}}^2=12\mathrm{x}\end{array}$
$point of intersection of the tangents to the parabola y^2=4ax at the points t_1 and t_2$
$\begin{array}{l}\mathrm{R}=\left({\mathrm{at}}_1{\mathrm{t}}_2, \mathrm{a}\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)\right)\\ \mathrm{\alpha }={\mathrm{at}}_1{\mathrm{t}}_2, \mathrm{\beta }=\mathrm{a}\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)\\ \frac{{\mathrm{\beta }}^2}{\mathrm{\alpha }}=\frac{{\mathrm{a}}^2{\left({\mathrm{t}}_1+{\mathrm{t}}_2\right)}^2}{a{t}_1{\mathrm{t}}_2}\\ =\frac{3{\left(3{\mathrm{t}}_2+{\mathrm{t}}_2\right)}^2}{\left(3{\mathrm{t}}_2\right){\mathrm{t}}_2}\\ \frac{3\times 16{\mathrm{t}}_2^2}{3{\mathrm{t}}_2^2}=16\end{array}$