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

Option: 1
$\mathrm{y^2=2 x}$

Option: 2
$\mathrm{x^2=2 y}$

Option: 3
$\mathrm{x^2=y}$

Option: 4
$\mathrm{y^2=x}$

Answers (1)

$\mathrm{O P: P Q=1: 3}$
Let the parametric co-ordinates of $\mathrm{Q\: be \: \left(4 t, 2 t^2\right)}$ We have, by section formula $\mathrm{\alpha=\frac{4 t+0}{4}=t\: and \: \beta=\frac{2 t^2+0}{4}=\frac{t^2}{2}}$ Eliminating $\mathrm{' t '}$ , we get the locus of $\mathrm{P(\alpha, \beta) \: as \: \alpha^2=2 \beta}$

Thus the locus is $\mathrm{x^2=2 y}$

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Kshitij

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