Question

Let $(x,y)$ be any point on the parabola $y^2=4x$. 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. $y^2=2x$
• B. $x^2=2y$
• C. $y^2=x$
• D. $x^2=y$

Answer 1

The correct option is C $y^2=x$


Let $P(\alpha ,\beta )$ be the point intersecting the line-segment joining $O(0,0)$ and $Q(t^2,2t)$ in the ratio $1:3$.
Then $\alpha =\frac{t^2+0}{4},\beta =\frac{2t+0}{4}$
$\Rightarrow \alpha =\frac{t^2}{4},\beta =\frac{t}{2}$
$\Rightarrow 4\alpha =(2\beta {)}^2$
$\Rightarrow {\beta }^2=\alpha$
Locus of $(\alpha ,\beta )$ is $y^2=x$