Question

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

• A. x = – a
• B. x = – a/2
• C. x = 0
• D. x = a/2

Answer 1

Answer: (C)

x = 0

Answer 2

The focus of the parabola y² = 4ax is S(a, 0).

Let P(at², 2at) be any point on the parabola. The

mid-point of SP is given by

x = $\frac{a(t^{2} + 1)}{2}$, y = $\frac{2at + 0}{2}$ = at

̃ 2x = a $\left\lbrack \frac{y^{2}}{a^{2}} + 1 \right\rbrack$ = $\frac{y^{2}}{a}$ + a

̃ y² = 2ax – a² ̃ y² = 2a $\left( x–\frac{a}{2} \right)$

̃ y² = 4AX {4A = 2a , A = a/2}

which is parabola with directrix X = – A

x – a/2 = – a/2 ̃ x = 0