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 is c. x = 0

Let P(at² , 2at) lies on the parabola

y² = 4ax

Mid point of PS is Q.

h = \(\frac{at^2+a}{2},K=\frac{0+2at}2\)

\(\frac{2h-a}a=t^2,\frac{K}a=t\)

\(\frac{2h-a}a=\frac{K^2}a\)

a(2h – a) = k²

Locus of (h, k) is y² =2a  x - a/2

Equaiton of directrix is \(x-\frac{a}2=-\frac{a}2\)

x = 0