Exam

JEE Main

Year

2026

Date

22 January

Shift

Evening Shift

Subject

Mathematics

Chapter

Parabola

JEE Main 2026 · 22 January · Evening Shift · Mathematics · Parabola

Let the locus of the mid-point of the chord through the origin O of...

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 :

Step-by-step solution

$y^2 = 4x$
Locus of mid point of OP
$M(h, k) \Rightarrow h = \dfrac{t^2}{2}, k = t$
$\Rightarrow k^2 = 2h \Rightarrow y^2 = 2x$

$S: y^2 = 2x$

$R(h, k)$
$\Rightarrow h = \dfrac{3t^2}{4}, k = \dfrac{3t}{4}$
$t^2 = \dfrac{8h}{3}, t = \dfrac{4k}{3}$
$\Rightarrow \dfrac{16k^2}{9} = \dfrac{8h}{3} \Rightarrow 2k^2 = 3h$
Locus of R : $2y^2 = 3x$