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/2`
C. `x = 0`
D. `x=a/2`

Answer

C

Explanation

Let `P(at^(2), 2at)` be a moving point on the parabola `y^(2)=4"ax and "S(a, 0)` be its focus. Let Q(h, k) be the mid-point of SP. Then,
`h=-(at^(2)+a)/(2)" and k"=(2at+a)/(2)`
`rArr" "(2h)/(a)=t^(2)+1" and "t=k/a`
`rArr" "(2h)/(a)=k^(2)/a^(2)+1rArrk^(2)=2ah-a^(2)" [On elimination t]"`
Thus, the locus of `(h, k) is y^(2)=2ax -a^(2)`
Now, `y^(2)=2ax -a^(2)rArr(y-0)^(2)=2a(x-a/2)`
The equation of the directrix of this parabola is
`x=-a/2=-a/2"i.e."x=0`.