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

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

Grade:11

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

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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