Question

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

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

Answer 1

The correct option is B $x=0$

Any point on the parabola $y^2=4ax$ be $(a{t}^2,2at)$
Let mid point of focus and variable point be $(h,k)$
$\therefore h=\frac{a{t}^2+a}{2},k=\frac{2at+0}{2}$
$\Rightarrow t^2=\frac{2h-a}{a}\text{and}t=\frac{k}{a}$
$\Rightarrow \frac{k^2}{a^2}=\frac{2h-a}{a}$
$\Rightarrow$ Locus of $(h,k)\text{is}{y}^2=a(2x-a)$
$\Rightarrow y^2=2a(x-\frac{a}{2})$
Its directrix is
$x-\frac{a}{2}=-\frac{a}{2}\Rightarrow x=0$