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The locus of the midpoint of the segment...

The locus of the midpoint of the segment joining the focus to a moving point on the parabola `y^2=4a x` is another parabola with directrix (a)`y=0` (b) `x=-a` (c)`x=0` (d) none of these

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The correct Answer is:
C
Let P,(h, k) be the mid-point of the line segment joining the focus (a,0) and a general point Q (,y) on the parabola, Then,
`h = (x + a)/(2), k = (y)/(2) implies x = 2k - a, y = 2k`,
Put these value of x and `y^(2) = 4ax`, we get
`4k^(2) = 4a (2h - a)`
`implies 4k^(2) = 8ah - 4a^(2) implies k^(2) = 2ah -a^(2)`
So, locus of P (h, K) is `y(2) = 2ax - a^(2)`
`implies y^(2) = 2a (x - (a)/(2))`
It directrix is `x - (a)/(2) = - (a)/(2) implies x = 0`
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