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Q.

The locus of the mid-point of the line segment joining the focus of the parabola $y^{2} = 4 a x$ to a moving point of the parabola, is another parabola whose focus is

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a

$(\frac{a}{2}, 0)$

b

$(a, 0)$

c

$(- \frac{a}{2}, 0)$

d

$(- a, 0)$

answer is B.

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Detailed Solution

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Here, $h = \frac{a t^{2} + a}{2} a n d k = \frac{2 a t + 0}{2}$
Question Image
$\Rightarrow t^{2} = \frac{2 h - a}{a} a n d t = \frac{k}{a}$
$∴ \frac{k^{2}}{a^{2}} = \frac{2 h - a}{a} \Rightarrow k^{2} = a (2 h - a)$
Locus of (h,k)is $y^{2} = a (2 x - a)$ .
$\Rightarrow y^{2} = 2 a (x - \frac{a}{2})$ , which is a parabola.
$∴ S = (a, 0)$

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