Q. show that the locus of the middle points of chords of parabola passing throughthe vertex is a parabola?

Hi Rohit,

Let the parabola we consider and draw chords be y² = 4ax.

The Vertex is O(0.0), which is one end of the chord. Let the other end be a varaible point P given by (at²,2at).

Let M(p,q) be the midpoint of the chord OP.

Midpoint of OP is (at²/2,at).

So, p = at²/2 and q = at. Now we have to eliminate "t" and get the relation between p and q to get the locus.

So t = q/a. Substitute this in the equation of p, and we will get

p = a/2*(q/a)²

So we have q² = 2ap.

Which is a parabola of the form y² = 2ax. And that proves the result.

Hope that helps.

All the best,

Regards,

Ashwin (IIT Madras).