The locus of the mid-point of the line segment joining the focus of the parabola y^2=4ax t…

Question

The locus of the mid-point of the line segment joining the focus of the parabola $y^2=4ax$ to a moving point of the parabola, is another parabola whose directrix is:

Accepted Answer

Midpoint of $(a,0)$ and $(at^2,2at)$: $h=\dfrac{at^2+a}2,\ k=at$. Then $t=\dfrac ka$, and
$$\frac{k^2}{a^2}=\frac{2h-a}a\ \Longrightarrow\ y^2=2a\left(x-\frac a2\right).$$

This has $4A'=2a\Rightarrow A'=\dfrac a2$; directrix at $x-\dfrac a2=-\dfrac a2\Rightarrow x=0$.

Solutions · 1