Solution
= Solution
Write $q(x,y)=x^2+y^2$. Equality of $q$-values defines an <equivalence relation induced by a function>: equality is <reflexive>, <symmetric>, and <transitive>, so all three properties hold for $R$. The <equivalence class> of $(x,y)$ is
$$
\boxed{[(x,y)]=\{(u,v):u^2+v^2=x^2+y^2\}.}
$$
For $(x,y)\ne(0,0)$ this is the <circle> centred at the origin with radius $\sqrt{x^2+y^2}$; the remaining <equivalence class> is the singleton $\{(0,0)\}$.