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Show that the locus of a point that divides a chord of slope 2 of a parabola y^2=4x internally in the ratio 1: 2 is a parabola. Find the vertex of the parabola.

Show that the locus of a point that divides a chord of slope 2 of a parabola y^2=4x internally in the ratio 1: 2 is a parabola. Find the vertex of the parabola.

Grade:12th pass

2 Answers

mycroft holmes
272 Points
9 years ago
Consider P(x) = (x-2)^2+(x-3)^2 This is the sum of two quadratics with real roots. But this quadratic is always positive and has no real roots.
noogler
489 Points
8 years ago
Let the two points on the given parabola be (t12, 2t1) and (t22, 2t2). Slope of the line joining these points is 
2=[2t2-2t1]/[t22-t12]=2/t1+t2

=> t1 + t2 = 1 

Hence the two points become (t12 , 2t1) and ((1 − t1)2 , 2(1 − t1))

Let (h , k) be the point which divides these points in the ratio 1 : 2
h=(1-t1)2+2t1/3=1-2t1+3t12/3       ....(1)
 

k=[2(1-t1)+4t1]/3=2+2t1/3               ....(2)
 

Eliminating t1 from (1) and (2), we find that 4h = 9k2 − 16k + 8 

Hence locus of (h , k) is 

(y-8/9)2=4/9(x-2/9)

This is a parabola with vertex (2/9 , 8/9)
 
 
 
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