Question

The locus of the midpoints of the line joining the focus and any point on the parabola $y^2=4ax$ is a parabola with the equation of directrix as.

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

Answer 1

Answer: (D)

$x=\frac{a}{2}$

Let a point lying on the parabola be $P=(h,k)$. The fous of the parabola $y^2=4ax$ is $F=(a,0)$.
Therefore the midpoint of the $PF=(\frac{h+a}{2},\frac{k}{2})$.
Hence $x=\frac{h+a}{2}$ and $y=\frac{k}{2}$.
Therefore
$h=2x-a$ and $k=2y$.
Now $k^2=4ah$ or $4y^2=4a(2x-a)$
$y^2=2ax-a^2$
Therefore the directrix of the parabola is
$2ax-a^2=0$ or $a(2x-a)=0$ since $a\ne 0$, $2x-a=0$ or $x=\frac{a}{2}$.