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

• A

  \(x^2=y\)
• B

  \(y^2=x\)
• C

  \(y^2=2 x\)
• Correct answer
  D

  \(x^2=2 y\)

Solution

Hint :
\(\begin{aligned} & O P: P Q=1: 3 \\ & \Rightarrow \mathrm{h}=\mathrm{t}, \mathrm{k}=\frac{\mathrm{t}^2}{2} \\ & \Rightarrow \mathrm{k}=\frac{\mathrm{h}^2}{2}\end{aligned}\)

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