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 locus of $$P$$ is :
A
$${y^2} = 2x$$
B
$${{x^2} = 2y}$$
C
$${{x^2} = y}$$
D
$${y^2} = x$$
Explanation
Let the coordinates of Q and P be (x1, y1) and (h, k) respectively.
$$\because$$ Q lies on x2 = 8y,
$$\therefore$$ x$$_1^2$$ = 8y ....... (1)
Again, P divides OQ internally in the ratio 1 : 3.
$$\therefore$$ $$h = {{{x_1} + 0} \over 4} = {{{x_1}} \over 4}$$ or x1 = 4h and