Question

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

• A. $(\frac{4}{9},\frac{8}{9})$
• B. $(\frac{2}{9},\frac{4}{9})$
• C. $(\frac{2}{9},\frac{8}{9})$
• D. $(\frac{4}{9},\frac{4}{9})$

Answer 1

The correct option is C $(\frac{2}{9},\frac{8}{9})$

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)