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

Options

A. x = -a
B. x = a
C. x = 0
D. x = a/2

Answer

C

Explanation

Let `P(at^(2), 2at)` be a moving point on the parabola `y^(2)=4ax` and let S(a, 0) be its focus.
Let Q(h, k) be the mid-point of PS. Then,
`rArr" "2h=a(k/a)^(2)+a" [On climinating t]"`
`rArr" "k^(2)=2a(h-a/2)`
Hence, the locus of `"(h, k) is "y^(2)=2a(x-a//2)`.
The equation of the directrix of this parabola is
`x-a/2=-a/2 rArr x=0`