Properties Of Sum Of Squares Relation

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Problem 63: (Properties Of Sum Of Squares Relation)

Let $A=\mathbb{R}\times\mathbb{R}$. Given $(a,b)\in A$ and $(x,y)\in A$, let's say that $(a,b)$ and $(x,y)$ are related by $R$ if and only if $a^2+b^2=x^2+y^2$.

1. Prove that $R$ is an equivalence relation.
2. What is the equivalence class of $(3,4)$? Plot the set of points in $ [(3,4)] $.
3. Given a point $(x,y)$, what shape will the equivalence class of $(x,y)$ be?

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