For a given parabola $y^2=4ax$ , two variable chords PQ and RS at right angle 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 $A(x_1,y_1)$ be PQ and RS
Equation of PQ is  $\frac{x-x_1}{\cos \theta }=\frac{y-y_1}{\sin \theta }=r$
Where $\tan \theta =$ slope of PQ
Any point on this line may be taken as
 $(x_1+r\cos \theta ,y_1+r\sin \theta )$
As the point lies on  $y^2=4ax$
 $\therefore {(y_1+r\sin \theta )}^2=4a(x_1+r\cos \theta )$
 $\Rightarrow r^2{\sin }^2+2y_1\sin \theta .r-4a\cos \theta r+y_1^2-4a{x}_1=0$
 $\therefore r_1{r}_2=\frac{(y_1^2-4a{x}_1)}{{\sin }^2\theta }$
Similarly $r_3{r}_4=\frac{(y_1^2-4a{x}_1)}{{\sin }^2\alpha }$ where $\tan \alpha =$ slope of RS
Since PQ is perpendicular to RS.
 $\therefore \alpha =90+\theta or\theta -90$
In either case  ${\sin }^2\alpha ={\cos }^2\theta$
Now $\frac{1}{r_1{r}_2}+\frac{1}{r_3{r}_4}=\frac{{\sin }^2\theta +{\cos }^2\theta }{(y_1^2-4a{x}_1)}=\frac{1}{(y_1^2-4a{x}_1)}$ which is constant