Question

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

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

Answer 1

The correct option is C $x=0$

Let $P(h,k)$ be the mid-point of the line segment joining the focus $(a,0)$ and a general point $Q(x,y)$ on the parabola. Then,
$h=\frac{x+a}{2},k=\frac{y}{2}\Rightarrow x=2h-a,y=2k$.
Put these values of $x$ and $y$ in $y^2=4ax$, we get
$4k^2=4a(2h-a)$
$\Rightarrow 4k^2=8ah-4q^2\Rightarrow k^2=2ah-a^2$
So, locus of $P(h,k)$ is $y^2=2ax-a^2$
$\Rightarrow y^2=2a(x-\frac{a}{2})$
Its directrix is $x-\frac{a}{2}=-\frac{a}{2}\Rightarrow x=0$