Parabola 1 Question 5
5. Let $O$ be the vertex and $Q$ 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
(2015)
(a) $x^{2}=y$
(c) $y^{2}=2 x$
(b) $y^{2}=x$
(d) $x^{2}=2 y$
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Answer:
Correct Answer: 5. (c)
Solution:
- PLAN Any point on the parabola $x^{2}=8 y$ is $\left(4 t, 2 t^{2}\right)$. Point $P$ divides the line segment joining of $O(0,0)$ and $Q\left(4 t, 2 t^{2}\right)$ in the ratio $1: 3$. Apply the section formula for internal division.
Equation of parabola is $x^{2}=8 y$
Let any point $Q$ on the parabola (i) is $\left(4 t, 2 t^{2}\right)$.
Let $P(h, k)$ be the point which divides the line segment joining $(0,0)$ and $\left(4 t, 2 t^{2}\right)$ in the ratio $1: 3$.
$$ \begin{aligned} & \therefore \quad h=\frac{1 \times 4 t+3 \times 0}{4} \Rightarrow h=t \\ & \text { and } \quad k=\frac{1 \times 2 t^{2}+3 \times 0}{4} \Rightarrow k=\frac{t^{2}}{2} \\ & \Rightarrow \quad k=\frac{1}{2} h^{2} \Rightarrow 2 k=h^{2} \\ & \Rightarrow \quad 2 y=x^{2} \text {, which is required locus. } \quad[\because t=h] \end{aligned} $$
