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  4. Let PQ be a double ordinate of the parabola, y2= - 4x, where P lies in the second quadrant. If R divides PQ in the ratio 2: 1, then the locus of R is :

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Q. 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 :

JEE MainJEE Main 2015Conic Sections

Solution:

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