Question

Find the locus of mid point of the focal chord of a parabola y^2=4ax
Please explain step by step

Answer 1

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.