Question

What is the equation of the locus of a point which moves such that 4 times its distance from the $x$-axis, is the square of its distance from the origin?

(A) $x^2+y^2-4 y=0$

(B) $x^2+y^2-4|y|=0$

(C) $x^2+y^2-4 x=0$

(D) $x^2+y^2-4|x|=0$

Answer 1

Let $(h, k)$ be the point.

$\begin{aligned}\because & 4 \sqrt{(h-h)^2+k^2} & =h^2+k^2 \\\Rightarrow & 4|k| & =h^2+k^2\end{aligned}$

Locus of the point is

$\begin{aligned}4|y| & =x^2+y^2 \\\Rightarrow \quad x^2+y^2-4|y| & =0\end{aligned}$

So, The correct option will be (B).