Question

Let $O$ be the vertex and $Q$ be any point on the parabola, $x^2=8y.$ If the point $P$ divide 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)

$x^2=2y$

The vertex of the Parabola is $O\equiv (0,0)$

And $Q(x_1,y_1)$ satisfies the parabola equation.

$\therefore x_1^2=8y_1$ ----(1)

Let $P\equiv (h,k)$

Given $P$ divides $OQ$ internally in ratio $1:3$

Thus $\frac{1\left(x_1\right)+3\left(0\right)}{1+3}=h$, $\frac{1\left(y_1\right)+3\left(0\right)}{1+3}=k$

$\frac{x_1}{4}=h$ and $\frac{y_1}{4}=k$

$x_1=4h$ and $y_1=4k$

Substituing this in (1)

$(4h{)}^2=8(4k)$

$h^2=2k$

i.e. $x^2=2y$