Proving Locus of Middle Points of Chords on Parabola

[Saitama]

Homework Statement

Prove that the locus of middle points of all chords of the parabola $y^2=4ax$ which are drawn through the vertex is the parabola $y^2=2ax$.

Homework Equations

The Attempt at a Solution

The mid-point of the line joining the vertex and a point $(h,2\sqrt{ah})$ on parabola is $(h/2,\sqrt{ah})$ but what am I supposed to do with this?

 

[I like Serena]

Homework Statement

Prove that the locus of middle points of all chords of the parabola $y^2=4ax$ which are drawn through the vertex is the parabola $y^2=2ax$.

Homework Equations

The Attempt at a Solution

The mid-point of the line joining the vertex and a point $(h,2\sqrt{ah})$ on parabola is $(h/2,\sqrt{ah})$ but what am I supposed to do with this?

You have $x=h/2$ and $y=\sqrt{ah}$. Eliminate h?

 

[Saitama]

You have $x=h/2$ and $y=\sqrt{ah}$. Eliminate h?

Woops, I was trying this question from the last night and I couldn't even notice that. Very foolish of me.

Thank you ILS!