Question

Let $PQ$ be a double ordinate of the parabola, $y^2= - 4x$, where P lies in the second quadrant. If R divides $PQ$ in the ratio $2 : 1$, then the locus of R is :

• A. $9y^2 = 4x$
• B. $9y^2 = - 4x$
• C. $3y^2 = 2x$
• D. $3y^2 = - 2x$

Answer 1

The Correct Option is B

$Let P\left(-at^{2}_{1}, 2at_{1}\right),$
$Q\left(-at^{2}_{1}, -2at_{1}\right)$ and $R\left(h, k\right)$
By using section formula, we have
$h=-at^{2}_{1}, k=\frac{-2at_{1}}{3}$
$k=-\frac{2at_{1}}{3}$
$\Rightarrow 3k=-2at_{1}$
$\Rightarrow 9k^{2}=4a^{2}\,t^{2}_{1}=4a\left(-h\right)$
$\Rightarrow 9k^{2}=-4ah$
$\Rightarrow 9k^{2}=-4h$
$\Rightarrow 9y^{2}=-4x$