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Q.
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y ^{2}=4 ax$ is another parabola with directrix
Conic Sections
Solution:
Let $P ( x , y )$ be the mid point of$ ( a , 0)$ and$\left( at ^{2}, 2 at \right)$
$2 x = a \left(1+ t ^{2}\right), y = at $
$ \therefore 2 x = a \left[1+\frac{y ^{2}}{a ^{2}}\right]= a +\frac{y ^{2}}{a} \Rightarrow y ^{2}=2 a \left( x -\frac{a}{2}\right)$
Vertex$ =\left(\frac{a}{2}, 0\right) ;$ Directrix $: x =\frac{a}{2}-\frac{a}{2}=0 \Rightarrow x =0 $
