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

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

Now I could solve this by the traditional method, and obtain the locus, and then find the directrix. Instead, I put the axis under consideration

The mid point of $$(0,0)$$ and $$(a,0)$$ is $$(\frac a2,0)$$, which is the vertex of the required parabola. Now will the focus scale down proportionally with parabola ie. will the focus still remain $$(a,0)$$, but with the length being $$\frac a2$$. If we assume that to be the case, the directrix will be $$x=0$$.

I don’t know whether the answer is right, but can anyone let me know if my think is correct?

Yes, it is OK. You can do this more formaly with a use of homothety. Remember that homothety preserves a shape (takes line to (paralel) line, circle to circle, ellipse to ellipse...). Now this new parabole is just a picture of starting one under a homothety with center at focus and dilatation factor $$k={1\over 2}$$ and you are done.