Prove that the locus at the middle points of the chords of the parabola y²=4ax which are drawn through the vertex is again a parabola y²=2ax.

Prove that the locus at the middle points of the chords of the parabola y²=4ax which are drawn through the vertex is again a parabola y²=2ax.

let, OP be the chord of parabola y² = 4ax.
let, locus of the middle point of the chord M(h , k).
let, P(x1,y1) be the any point on the parabola.

so, y1² = 4ax1 ---------------- ( I )

Middle point of OP= ( (x1+0)/2 , (y1+0)/2 )
or, (h , k) = ( x1/2 , y1/2 )
or, h = x1/2 , k = y1/2
or, x1 = 2h , y1 = 2k

From equation( I ),
(2k)² = 4a(2h)
or, 4k² = 8ah
o, k² = 2ah

Since, (h , k) be locus.
so, y² = 2ax.

Hence, proved.