Question

The locus of the mid-point of the line segment joining the focus of the parabola $y^2=4$ ax to a moving point of the parabola, is another parabola whose directrix is :

• A. $x=-\frac{a}{2}$
• B. $x=\frac{a}{2}$
• C. $x=0$
• D. $x=a$

Answer 1

Answer: (C)

$h=\frac{a{t}^2+a}{2},k=\frac{2at+0}{2}$
$\Rightarrow t^2=\frac{2h-a}{a}$ and $t=\frac{k}{a}$
$\Rightarrow \frac{k^2}{a^2}=\frac{2h-a}{a}$
$\Rightarrow$ Locus of $(h,k)$ is $y^2=a(2x-a)$
$\Rightarrow y^2=2a\left(x-\frac{a}{2}\right)$
Its directrix is $x-\frac{a}{2}=-\frac{a}{2}\Rightarrow x=0$