Question

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

(a)  y = 0

(b)  x = –a 

(c)  x = 0

(d)  none of these

Answer 1

Correct option  (c)  x = 0

Explanation:

Let P(at² , 2at) lies on the parabola

y² = 4ax

Mid point of PS is Q.

h = at²  + a/2, k = 0 + 2at/2

2h – a/a = t², k/a = t

2h – a/a = (k/a)²

a(2h – a) = k²

Locus of (h, k) is

y² = 2a(x - a/2)

Equaiton of directrix is x -a/2 = -a/2

 x = 0