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 whose directrix is $x=k$. The value of $k$ is

Answer 1

Focus of the parabola $y^2=4ax$ is $S(a,0)$
And point $P$ on the parabola is taken as $(a{t}^2,2at)$
Now, mid point of SP is $(\frac{a+a{t}^2}{2},at)=R(h,k)$, say
$\Rightarrow a(1+t^2)=2h$ and $k=at\Rightarrow t=k/a$
Now eliminating $t$ we get, $a(1+k^2/a^2)=2h$
$\Rightarrow k^2=2ah-a^2$
Thus locus of R is $y^2=2ax-a^2$
$y^2=2a(x-a/2)$
Thus directrix of this parabola is, $x-a/2=-\left(a/2\right)\Rightarrow x=0$