Let O be the vertex and Q be any point on the parabola `x^2=8y`. IF the point P divides the line segment OQ internally in the ratio 1:3 , then the locus of P is

Options

A. `y^(2)=2x`
B. `x^(2)=2y`
C. `x^(2)=y`
D. `y^(2)=x`

Answer

B

Explanation

Let the parametric coordinates of Q be `(4t, 2t^(2))` and let the coordinates of P be (h, k).
It is given that P divides OQ in the ratio 1 : 3
`:." "h=(1xx4t+3xx)/(1+3)" and "k=(1xx2t^(2)+3xx0)/(1+3)`
`rArr" "h=t " and "k=t^(2)/2rArrl=h^(2)/2rArrh^(2)=2k`
Hence, the locus P is `x^(2)=2y`.