Let O be the vertex and Q be any point on the parabola x^2… — JEE Main 2015 Mathematics PYQ

JEE Main 2015

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

A.
$x^2 = y$
B.
$y^2 = x$
C.
$y^2 = 2x$
D.
$x^2 = 2y$

Show answer and solution

Correct option: D

Solution

Q is on x²=8y, so parametrize Q as (4t, 2t²). O is (0,0). P divides OQ in ratio 1:3, so P = (1·Q + 3·O)/(1+3) = (4t/4, 2t²/4) = (t, t²/2). Let P=(h,k): h=t, k=t²/2, so k=h²/2, giving h²=2k. The locus is x²=2y.

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JEE Main 2015 Mathematics question on Conic Sections