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Q.

Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . Let P be the point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $2 : 3$ . Then locus of $P$ is

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a

$5 x^{2} = 2 y$

b

$x^{2} = y$

c

$5 y^{2} = 8 x$

d

$5 y^{2} = 2 x$

answer is C.

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Detailed Solution

Let $P$ be $(h, k)$

then $h = \frac{2 x}{5}, k = \frac{2 y}{5}$

As $(x, y)$ lies on the parabola $y^{2} = 4 x$

${(\frac{5 k}{2})}^{2} = 4 (\frac{5 h}{2})$

$\Rightarrow 5 k^{2} = 8 h$

$\Rightarrow$ Locus of $P$ is $5 y^{2} = 8 x$ .

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