Question

A pair of variable straight lines $5 x^2+3 y^2+\alpha x y=0(\alpha \in R)$, 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{ yk -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
$5 x^2+3 y^2+\alpha x y=0$ ....(2)

As (1) and (2) are same
$\Rightarrow \frac{8}{5}=\frac{2 h }{3}=-\frac{4 k }{\alpha}$
$\Rightarrow $ Required locus is $x=\frac{12}{5}$
$\left[\right.$ From $\frac{8}{5}=\frac{2 h}{3}$