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)

$x^{2}=2y$

Answer 2

Let $P:(h, k)$
$h=\frac{1 . \alpha+\beta .0}{4}$
$\Rightarrow \alpha=4 h$
$k=\frac{1 . \beta+3.0}{4}$
$\Rightarrow \beta=4 k$
$\because(\alpha, \beta)$ on Parabola
$\Rightarrow \alpha^{2}=8 \beta$
$\Rightarrow\left(4 h^{2}\right)=8.4 k$
$16 h^{2}=32 k$
$x^{2}=2 y$