Equivalence classes on $\,\Bbb R^2.$

Question

I'm really confused about the equivalence class of the following relation:

On $\,\Bbb R^2,\;xRy\;$ if $\;|x|=|y|\;$ where $\,|x|=\sqrt{x_1^2+x_2^2}\,.$

Answer 1

Two points are related if they are if their distance to the origin is the same. That's clearly an equivalence relationship and the classes are the collection of circles centered at the origin with radius $r>0$ and the point $(0, 0) $, that we can consider as a circle of radius $r=0$, so $$\mathcal{R}=\{\{x^2+y^2=r^2\}:r\geq 0\}$$