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

Answer 1

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

Let the coordinates of $P$, whose locus is to be determined to be $(h,k)$.

Since this point divides the line joining $(0,0)$ to $(x,y)$ in the ratio $1:3$. So coordinates of point $P$ will be $(\frac{x}{4},\frac{y}{4})$

So, $h=\frac{x}{4}$ and $k=\frac{y}{4}$

Or, $x=4h$, and $y=4k$

Since, $y^2=4x$

$\Rightarrow$ $k^2=h$

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