Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$\alpha =\frac{a{t}^2+a}{2},\beta =\frac{2at+0}{2}\Rightarrow 2\alpha =a{t}^2+at=\beta$
$\therefore 2\alpha =a.\frac{{\beta }^2}{a^2}+a$ or $2a\alpha ={\beta }^2+a^2$
$\therefore$ The locus is $y^2=\frac{4a}{2}(x-\frac{a}{2})=4b(x-b),(b=\frac{a}{2})$
$\therefore$ Directrix is (x-b)+b=0 or x=0.