Question

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

• A. $x^2=y$
• B. $y^2=x$
• C. $y^2=2x$
• D. $x^2=2y$

Answer 1

Answer: (D)

$x^2=2y$

Vertex of the parabola $x^2=8y$ is $O(0,0)$

And any point on the parabola is $Q(4t,2t^2)$

Let $P(h,k)$ divides the line segment OQ internally in the ratio $1:3$

$\Rightarrow h=\frac{3\times 0+1\times 4t}{1+3}=t$

and $k=\frac{3\times 0+1\times 2t^2}{1+3}=\frac{t^2}{2}$

Eliminating $t$ we get, $h^2=2k$

Hence the locus of $p(h,k)$ is, $x^2=2y$