First slide

Parabola

Question

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 a x$ is another parabola with directrix

Moderate

A

$x = - a$
B

$x = - a / 2$
C

$x = 0$
D

$x = a / 2$

Solution

The focus of the parabola $y^{2} = 4 a x$ is $S (a, 0)$ , let $P (a t^{2}, 2 a t)$ be any point on the parabola then coordinates of the mid-point of $S P$ are given by
$x = \frac{a (t^{2} + 1)}{2}, y = \frac{2 a t + 0}{2}$
Eliminating $‘ t ’$ we get the locus of the mid-point
as $y^{2} = 2 a x - a^{2}$ or $y^{2} = 2 a (x - a / 2)$ (1)
which is a parabola of the form $Y^{2} = 4 A X$ (2)
where $Y = y, X = x - a / 2$ and $A = a / 2$
Equation of the directrix of (2) is $X = - A$
So equation the directrix of (1) is $x - a / 2 = - a / 2$
$\Rightarrow x = 0$

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