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  4. Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1: 3, then the locus of P is

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Q. 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

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Solution:

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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 $
$\Rightarrow \left(4 h^{2}\right)=8.4 k$
$16 h^{2}=32 k$
$x^{2}=2 y$