Is it true that $\langle x^2+y^2-1\rangle$ is prime in $\mathbb{C}[x,y]$? Is it true that $\langle x^2+y^2-1\rangle$ is prime in $\mathbb{C}[x,y]$ ?
One way to see that this ideal is prime is to show that the quotient ring is an integral domain. By change of variables $u=x+iy, v=x-iy$ you get that $$\mathbb{C}[x,y]/\left&lt;x^2+y^2-1\right&gt;\cong \mathbb{C}[u,v]/\left&lt;uv-1\right&gt;\cong \mathbb{C}[t^{\pm 1}].$$
Yes. Since the  conic $x^2+y^2=1$ is  an irreducible curve in $\mathbb{C}^2$.

More than irreducible, its projective closure is smooth. One can simply check the vanishing of the partial derivatives.