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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).
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