Question

A pair of variable straight lines $5x^2+3y^2+\alpha xy=0(\alpha \in R)$, cut $y^2=4x$ 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

Answer: 17

Let point of intersection of tangents be $(h,k)$ chord of contact is $PQ\equiv yk=2(x+h)$
$\Rightarrow$ joint equation of $OP$ and $OQ$ where $O$ is origin.
$\Rightarrow y^2-4x\left[\frac{yk-2x}{2h}\right]=0$
$\Rightarrow 2h{y}^2-4kxy+8x^2=0$....(1)
Also from question joint equation of $OP$ and $OQ$ is
$5x^2+3y^2+\alpha xy=0$ ....(2)

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