Equivalence class of $T$ on $\mathbb{R} \times \mathbb{R}$ given by $(x,y) T (a,b)$ iff $x^{2}+y^{2}=a^{2}+b^{2}$

Question

What is the equivalence class of $T$ on $\mathbb{R} \times \mathbb{R}$ given by $(x,y) T (a,b)$ iff $x^{2}+y^{2}=a^{2}+b^{2}$

I can see that the equivalence class cannot be negative, as the square of any real number is positive.

So is the equivalence class simply $\left \{ \mathbb{\left \{R \right \}}> 0 \right \}$?

But I think this is incorrect since the equivalence class entails nothing (except the answer is always positive) about the relation T. Can someone please help me?

Answer 1

The equivalence classES of $T$ consists of all circles that are centered at the origin. $(x, y)T (a, b)$ then means that $(x, y)$ lies on the same circle as $(a, b)$, as do all points lying on that given circle of radius $r$. For each $r\in \mathbb R_{\geq 0}$, there is one and only one equivalence class, that being the set of points that lie on the circle $$x^2 + y^2 = r^2$$