QJEEM15Ma2e5b: JEE Main 2015 Mathematics - Class 11 JEEM15Ma2e5b Solution: Let be the vertex and be any point on the parabola, . If the…

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

$k = {{{y_1} + 0} \over 4} = {{{y_1}} \over 4}$ or y₁ = 4k

Now putting x₁ and y₁ in (1) we get,

16h² = 32k or, h² = 2k

$\therefore$ the locus of P is given by, x² = 2y.