Question

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

• A. $x^2=y$
• B. $y^2=2xy$
• C. $y^2=x$
• D. $x^2=2y$

Answer 1

The correct option is C

$y^2=x$

Explanation for the correct option.

Let the coordinates of $P$ be $\left(h,k\right)$. It divides the line segment from $(0,0)\mathrm{to}(x,y)$ in the ratio $1:3$, so

$\begin{array}{rcl}h& =& \frac{x}{4}\mathrm{and}k=\frac{y}{4}\\ & \Rightarrow & x=4h\mathrm{and}y=4k\\ & & \end{array}$

Now, $y^2=4x$

$\begin{array}{rcl}& \Rightarrow & {\left(4k\right)}^2=4\left(4h\right)\\ \Rightarrow k^2& =& h\\ \Rightarrow y^2& =& x\end{array}$

So, the locus of $P$ is $y^2=x$.

Hence, option C is correct.