Show that the locus of a point that divides a chord of slope 2 of y2=4x internally in the ratio 1:2 is a parabola. Also find its vertex.
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)