The equation to the locus of a point which moves so that its distance from $x$-axis is always one half its distance from the origin, is

The correct option is B. $x^2-3y^2=0$

Let the moving point be $P(x,y)$ and its distance from $x$- axis is y.

Therefore, according to given condition

$\frac{1}{2}\sqrt{x^2+y^2}=y$ [Since, distance of a point from the origin is $\sqrt{x^2+y^2}$]

$\Rightarrow \frac{1}{4}(x^2+y^2)=y^2$

$\Rightarrow x^2+y^2=4y^2$

$\Rightarrow x^2+y^2-4y^2=0$

$\therefore$ $x^2-3y^2=0$ is the required locus.