Question

A pair of variable straight lines $5x^2+3y^2+\alpha xy=0(\alpha \in R)$, cut the parabola $y^2=4x$ at two points (other than origin) $P$ and $Q$. If the locus of the point of intersection of tangents to the given parabola at $P$ and $Q$ is given by $x=\frac{p}{q}$ (where $p,q$ are natural numbers coprime to each other), then the value of $(p–q+2)$ is equal to

• A. 6
• B. 9
• C. 4
• D. 5

Answer 1

The correct option is B 9


Let the point of intersection of tangents be $R(h,k)$
\therefore Equation of chord of contact $PQ$ is
$ky=2(x+h)$
$\therefore$ Join equation of $OP$ and $OQ$ having $‘Q^{\prime }O$ as the origin is given by
$y^2-4x\cdot \left(\frac{yk-2x}{2h}\right)=0$
$\Rightarrow 2h{y}^2-4kxy+8x^2=0....\left(1\right)$
Which is identical with the equation
$5x^2+3y^2+\alpha xy=0...\left(2\right)$
Comparing (1) and (2)
$\frac{8}{5}=\frac{2h}{3}=\frac{-4k}{\alpha }$
$\Rightarrow 2h=\frac{3\times 8}{5}$
$\therefore LocuofR(h,k)isx=\frac{12}{5}=\frac{p}{q}\left(given\right)$
$\therefore p-q+2=12-5+2=9$