Question

The locus of the mid-point of the line segment joining the focus ot 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

$x=0$

$S=(a,0),P(a{t}^2,2at).\text{If}(h,k)$ be mid-point, then
$2h=a(1+t^2),2k=2at\text{or}k=at$
$\therefore 2x=a\left(1+\frac{y^2}{a^2}\right)\text{or}{y}^2=2a\left(x-\frac{a}{2}\right)\text{or}{Y}^2=4AX$
$\therefore \text{Directrix is}X=-A\text{or}x-\frac{a}{2}=-\frac{2a}{4}\text{or}x=0.$