For a given parabola  $y^2=4ax$, two variable chords  $PQ$  and  $RS$ at right angles are drawn through the fixed point  $A(x_1,y_1)$ inside the parabola, making variable angles  $\theta$ and  $\alpha$  with x-axis. If  $r_1,r_2,r_3,r_4$ are distances of  $P,Q,R$  and  S from  A, then the value of  $\frac{1}{r_1{r}_2}+\frac{1}{r_3{r}_4}$

Let two perpendicular chords through  $\text{A}({\text{x}}_1,{\text{y}}_1)$ be  PQ and  RS
Equation of  PQ is  $\frac{x-x_1}{\text{cos}\theta }=\frac{y-y_1}{\text{sin}\theta }=r$
Where $\text{tan}\theta =$ slope of  PQ
An point on this line may be taken as  $(x_1+r\text{cos}\theta ,y_1+r\text{sin}\theta )$
As the point lies on,
$\therefore {(y_1+r\text{sin}\theta )}^2=4a(x_1+r\text{cos}\theta )\Rightarrow r^2{\text{sin}}^2\theta +2y_1\text{sin}\theta r-4a\text{cos}\theta r+y_1^2-4a{x}_1=0$

$$\begin{gathered} \therefore r_1{r}_2=\frac{(y_1^2-4a{x}_1)}{{\text{sin}}^2\theta } \\ \text{Similarly }{r}_3{r}_4=\frac{(y_1^2-4a{x}_1)}{{\text{sin}}^2\alpha } \\ \text{Where tan }\alpha =\text{ slope of RS } \\ \text{Since PQ is perpendicular to RS, } \end{gathered}$$

Similarly  $r_3{r}_4=\frac{(y_1^2-4a{x}_1)}{{\text{sin}}^2\alpha }$
Where $\text{tan}\alpha =$ slope of RS
Since PQ is perpendicular to RS,    $\therefore \alpha =90+\theta \text{ or }\theta -90$