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# The locus of the mid point of the line segment joining the focus to a moving point on parabola $y^2=4ax$ is another parabola with direction

$\begin{array}{1 1}(A)\;x=-a \\(B)\;x=\large\frac{-a}{2} \\(C)\;x=0 \\(D)\;x=\large\frac{a}{2} \end{array}$

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## 1 Answer

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Take any point on the parabola $(at^2,2at)$
Therefore the mid point $(h,k)=\bigg(\large\frac{at^2+a}{2},\frac{2at+0}{2}\bigg)$
$2h=a(t^2+1),at=k$
$2h= a (\large\frac{k^2}{a^2}+1)$
So,locus is $2xa=y^2+a^2$
$y^2=2a(x-a/2)$
Direction : x $=\large\frac{-a}{2}+\frac{a}{2}$
$\qquad=0$ or y axis
Hence C is the correct answer.
answered Apr 10, 2014 by

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