Let O be the vertex and Q any point on the parabola x^2=8y. If point P divides segment OQ internally in the ratio 1:3, find the locus of P.

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EEJ MAIN Mathematics QUESTION #6044

Question 1

Let $O$ be the vertex and $Q$ any point on the parabola $x^2=8y$. If point $P$ divides segment $OQ$ internally in the ratio $1:3$, find the locus of $P$.

$x^2=y$
$y^2=x$
$y^2=2x$
$x^2=2y$✔️

Correct Answer Explanation

Let $Q=(4t, 2t^2)$ be a point on $x^2=8y$ (using parametrization $x=4t, y=2t^2$). $O=(0,0)$.

$P$ divides $OQ$ in ratio $1:3$: $P=\left(\frac{1\cdot4t+3\cdot0}{4},\frac{1\cdot2t^2+3\cdot0}{4}\right)=\left(t,\frac{t^2}{2}\right)$

Let $P=(h,k)$: $h=t$ and $k=\frac{t^2}{2}=\frac{h^2}{2}$

So $h^2=2k$, i.e., the locus is $x^2=2y$.

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