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 :

• A

  $2 x^{2}=3 y$
• B

  $3 y^{2}=2 x$
• Correct answer
  C

  $2 y^{2}=3 x$
• D

  $3 x^{2}=2 y$

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 : $2y^2 = 3x$

Hence, the correct answer is option: 'C'.