Two tangents  $2x+y=2$ and $x-2y=3$  to a parabola touching it at  $A(2,-2)$ and$B(5,1)$   respectively. If focus of parabola is  $S(\alpha ,\beta )$ and length of latus rectum is L then which of the following option(s) is/are TRUE?  $(\alpha ,\beta \in R)$

As tangents are perpendicular to each other and its points of intersections $P(\frac{7}{5},-\frac{4}{5})$  lies on directrix. AB is focal chord.
Equation of AB is x – y =4, as focus is  $(\alpha ,\beta )$
so  $\alpha -\beta =4\Rightarrow \beta =\alpha -4$
Now  $\angle ASP={90}^{\circ }$
 ${ }_{\Rightarrow \frac{-2-\alpha +4}{2-\alpha }\cdot \frac{-\frac{4}{5}-\alpha +4}{\frac{7}{5}-\alpha }=-1}$
 ${ }_{\Rightarrow \frac{2-\alpha }{2-\alpha }\cdot \frac{\frac{16}{5}-\alpha }{\frac{7}{5}-\alpha }=-1}$
 $\Rightarrow \frac{16}{5}-\alpha =\alpha -\frac{7}{5};\alpha =\frac{23}{10}$
 $\Rightarrow$ Focus is  $(\frac{23}{10},-\frac{17}{10})$
Also,  ${\mathcal{l}}_1=AS=\frac{3\sqrt{2}}{10},{\mathcal{l}}_2=BS=\frac{27\sqrt{2}}{10}$
 $\frac{1}{a}=\frac{1}{{\mathcal{l}}_1}+\frac{1}{{\mathcal{l}}_2}\Rightarrow a=\frac{27\sqrt{2}}{100}\Rightarrow 4a=\frac{27\sqrt{2}}{25}$