If $P_1{P}_2$ and $Q_1{Q}_2$ two focal chords of a parabola are at a right angles, then:

${\mathrm{P}}_1=\left({\mathrm{at}}_1^2,2{\mathrm{at}}_1\right)$ s = (a, 0)
Let, ${\overline{\mathrm{SP}}}_1$ make angle $\mathrm{\theta }$ with axis
$\begin{array}{l}\Rightarrow \text{ slope of }\overline{{\mathrm{SP}}_1}=\tan \mathrm{\theta }\\ \Rightarrow \frac{2{\mathrm{t}}_1}{{\mathrm{t}}_1^2-1}=\tan \mathrm{\theta }\Rightarrow {\mathrm{t}}_1-\frac{1}{{\mathrm{t}}_1}=2\cot \mathrm{\theta }\\ \therefore {\left({\mathrm{t}}_1+\frac{1}{\mathrm{t}}\right)}^2={\left({\mathrm{t}}_1-\frac{1}{{\mathrm{t}}_1}\right)}^2+4=4{\mathrm{cosec}}^2\mathrm{\theta }\end{array}$
$\Rightarrow$ P₁P₂ length of focal chord.
$$\begin{gathered} =\mathrm{a}{\left({\mathrm{t}}_1+\frac{1}{{\mathrm{t}}_1}\right)}^2=4{\mathrm{cosec}}^2\mathrm{\theta } \\ \text{ Let }{\mathrm{Q}}_1=\left({\mathrm{at}}_2^2,2{\mathrm{at}}_2\right)\text{. } \end{gathered}$$
Let ${\overline{\mathrm{SQ}}}_1$ make $\mathrm{\theta }+\frac{\mathrm{\pi }}{2}$ with axis.
 slope of ${\overline{\mathrm{SQ}}}_1=\tan \left(\frac{\mathrm{\pi }}{2}+\mathrm{\theta }\right)$
Similarly ${\mathrm{Q}}_1{\mathrm{Q}}_2=4{\mathrm{cosec}}^2\left(\frac{\mathrm{\pi }}{2}+\mathrm{\theta }\right)=4{\sec }^2\mathrm{\theta }$
Area of quadrilateral ${\mathrm{P}}_1{\mathrm{Q}}_1{\mathrm{P}}_2{\mathrm{Q}}_2\text{ is }\frac{1}{2}\overline{{\mathrm{P}}_1{\mathrm{P}}_2}\times {\mathrm{Q}}_1{\mathrm{Q}}_2=\frac{1}{2}\left({\mathrm{P}}_1{\mathrm{P}}_2\right)\left({\mathrm{Q}}_1{\mathrm{Q}}_2\right)\left(\sin \frac{\mathrm{\pi }}{2}\right)$
$$\begin{gathered} =\frac{1}{2}\cdot 4{\mathrm{cosec}}^2\mathrm{\theta }\cdot 4{\sec }^2\mathrm{\theta }. \\ =\frac{8}{{\sin }^2{\mathrm{\theta cos}}^2\mathrm{\theta }}=\frac{32}{{\sin }^{2}2\mathrm{\theta }}. \end{gathered}$$
Area is minimum when sin²2$\mathrm{\theta }$ = 1
$\Rightarrow 0=\frac{\mathrm{\pi }}{4}.$
Also min. area = 32