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Q.

The locus of the midpoint 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 directrix equation is

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a

x=0

b

x=a/2

c

x=a

d

x=-a/2

answer is B.

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Detailed Solution

$\begin{array}{l} S (a, 0), P (a t^{2}, 2 a t) \Rightarrow mid point (x_{1}, y_{1}) = (a (\frac{1 + t^{2}}{2}) \cdot a t) \\ t = \frac{y_{1}}{a}, 2 x_{1} = a + a t^{2} = a + 4 \cdot \frac{y_{1}^{2}}{a} \\ \Rightarrow y_{1}^{2} = 2 a x_{1} - a^{2} \Rightarrow y_{1}^{2} = 2 a (x_{1} - \frac{a}{2}) \end{array}$
$Locus is y^{2} = 2 a (x - \frac{a}{2}) \Rightarrow its direcrix is x - \frac{a}{2} + \frac{a}{2} = 0 \Rightarrow x = 0$

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