Proving a Certain $\mathbb{C}$-Algebra is a Domain Using a Specified Method
Since the question contains a mistake, let me fix it.
The ring $R=\mathbb{C}[X,Y]/(X^2+Y^2-1)$ is a UFD because it is isomorphic to $\mathbb{C}[U,V]/(UV-1)$ (via the substitutions $U\mapsto X+iY$ and $V\mapsto X-iY$) and the last one is a ring of fractions of $\mathbb{C}[U]$ (with respect to the multiplicative system $\{1,U,U^2,\dots\}$).
Remark. If $R$ is an integral domain and $\alpha\in R$, then $R[X]/(X^2−\alpha)$ is an integral domain if and only if there are no non-zero elements $a,b\in R$ such that $b^2=\alpha a^2$.