Let $(x, y)$ be any point on the parabola $y^2=4 x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x, y)$ in the ratio $1: 3$. Then, the locus of $P$ is

• A

  $x^2=y$
• B

  $y^2=2 x$
• Correct answer
  C

  $y^2=x$
• D

  $x^2=2 y$

Solution

$$ \text { By section formula, } $$
$h=\frac{x+0}{4}, k=\frac{y+0}{4}$ $\therefore \quad x=4 h$ and $y=4 k$ Substituting in $y^2=4 x$, $(4 k)^2=4(4 h) \Rightarrow k^2=h$ or $y^2=x$ is required locus.

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