Let the locus of the mid-point of the chord through the origin O of the parabola y 2 =4 x be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio 3: 1 , is :

Mathematics · Parabola

JEE Main 2026 — 22 January, Evening Shift — Question 17

Let the locus of the mid-point of the chord through the origin O of the parabola $\mathrm{y}^{2}=4 \mathrm{x}$ be the curve S . Let P be any point on S . Then the locus of the point, which internally divides OP in the ratio $3: 1$ , is :

Option A:
$3 y^{2}=2 x$
Option B:
$2 y^{2}=3 x$
Correct
Option C:
$3 x^{2}=2 y$
Option D:
$2 x^{2}=3 y$

Answer: B

Step-by-step solution

$y^{2}=4 x$

Locus of mid point of OP $\mathrm{M}(\mathrm{h}, \mathrm{k}) \Rightarrow \mathrm{h}=\frac{\mathrm{t}^{2}}{2}, \mathrm{k}=\mathrm{t}$

$\Rightarrow \mathrm{k}^{2}=2 \mathrm{~h} \Rightarrow \mathrm{y}^{2}=2 \mathrm{x}$

$\mathrm{S}: \mathrm{y}^{2}=2 \mathrm{x}$

R(h, k) $\Rightarrow \mathrm{h}=\frac{\frac{3 \mathrm{t}^{2}}{2}}{4}, \mathrm{k}=\frac{3 \mathrm{t}}{4}$

$\mathrm{t}^{2}=\frac{8 \mathrm{~h}}{3}, \mathrm{t}=\frac{4 \mathrm{k}}{3}$

$\Rightarrow \frac{16 \mathrm{k}^{2}}{9}=\frac{8 \mathrm{~h}}{3} \Rightarrow 2 \mathrm{k}^{2}=3 \mathrm{~h}$

Locus of R : $2 \mathrm{y}^{2}=3 \mathrm{x}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 22 January, Evening Shift
Subject: Mathematics
Chapter: Parabola
Topic: Introduction to Parabola