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

Answer

D

Explanation

Any point on the parabola `x^(2) = 6y` is `(4 t, 2t^(2))`. Point divides the line segment joining of O(0,0) and `Q (4t , 2t)^(2)` in the ratio 1 : 2. Apply the section formula for internal division.
Equation of parabola is `x^(2) = 8y`
Let any poin Q on the parabola is `(4 t, 2 t^(2))`
Let P (h, k) be the point which divides the line segment joining (0,0) and `(4 t, 2t^(2))` in the ratio 1 : 3.

`:. h = (1 xx 4t + 3 xx 0)/(4) implies h = t`
and `k = (1 xx 2t^(2) + 3 xx 0)/(4) implies k = (t^(2))/(2)`
`implies k = (1)/(2) h^(2) implies 2k = h^(2)` `[:' t = h]`
`implies 2y = x^(2)`, which is required locus,