Question

A pair of variable straight lines $5 x ^2+3 y ^2+\alpha xy =0(\alpha \in R )$, cut $y ^2=4 x$ at two points $P$ and $Q$. If the locus of the point of intersection of tangents to the given parabola at $P$ and $Q$ is $x=\frac{m}{n}$ (where $m$ and $n$ are in their lowest form), find the value of $(m+n)$.

Answer 1

Let point of intersection of tangents be $(h, k)$
chord of contact is $P Q \equiv y k=2(x+h)$
$\Rightarrow $ joint equation of $O P$ and $O Q$ where $O$ is origin.

$\Rightarrow y^2-4 x\left[\frac{y k-2 x}{2 h}\right]=0$
$\Rightarrow 2 hy ^2-4 kxy +8 x ^2=0$ ......(1)
Also from question joint equation of $O P$ and $O Q$ is
$5 x ^2+3 y ^2+\alpha xy =0$ ..... (2)
As (1) and (2) are same
$\Rightarrow \frac{8}{5}=\frac{2 h }{3}=-\frac{4 k }{\alpha}$
$\Rightarrow $ Required locus is $x =\frac{12}{5}$
[From $\frac{8}{5}=\frac{2 h }{3}$ ]