The locus of the midpoint of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 a x$ , is another parabola with the directrix

x = \frac{a}{2}

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x = - \frac{a}{2}

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x = 0

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x = a

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Solution

The correct option is C $x = - \frac{a}{2}$
Assume any point on the given parabola is $P (a t^{2}, 2 a t)$ and its focus is $S (a, 0)$
So mid point of $P S$ is $(\frac{a + a t^{2}}{2}, a t)$
$\Rightarrow x = \frac{a + a t^{2}}{2}, y = a t$
Eliminating $t$ we get required locus $y^{2} = 2 a (x - \frac{a}{2})$ which has directrix $x = - \frac{a}{2}$

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The locus of the midpoint of the line segment joining the focus to a moving point on the parabola y2=4ax, is another parabola with the directrix

The correct option is C x=−a2Assume any point on the given parabola is P(at2,2at) and its focus is S(a,0)So mid point of PS is (a+at22,at)⇒x=a+at22,y=atEliminating t we get required locus y2=2a(x−a2) which has directrix x=−a2

The locus of the midpoint of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 a x$ , is another parabola with the directrix