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

JEE Main · 2015 · 04 Apr 2015 · Offline · Parabola

Let O be the vertex and Q be any point on the parabola,

x^{2} = 8 y

. If the point P divides the line segment OQ internally in the ratio

1 : 3

, then the locus of P is :

(A) $y^{2} = x$
(B) $y^{2} = 2 x$
(C) $x^{2} = 2 y$ ✓ Correct
(D) $x^{2} = y$

Answer: (C)

Solution

Take

Q = (h, k)

on the parabola, so

h^{2} = 8 k

. With the vertex

O

at the origin, the point

P

dividing

O Q

1 : 3

lands at

(\frac{h}{4}, \frac{k}{4})

. Writing

P = (x, y)

gives

h = 4 x

k = 4 y

, and substituting into

h^{2} = 8 k

(4 x)^{2} = 8 (4 y)

, i.e.

16 x^{2} = 32 y

, which tidies to

x^{2} = 2 y

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option

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