Question

Prove that the locus of the middle points of all chords of the parabola

Answer 1

Solution

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=at22
and
q=at
Now we have to eliminate t and get the relation between p and q to get the locus.

So,

t=qa
Substitute this in the equation of p, and we will get:

p=a2(qa)2
So we have,
q2=2ap
Which is a parabola of the form y2=2ax

Answer 2

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=at22

andq=at

Now we have to eliminate t and get the relation between p and q to get the locus.

So,

t=qa

Substitute this in the equation of p, and we will get:

p=a2(qa)2

So we have,q2=2ap

Which is a parabola of the form y2=2ax