How to find the quotient $\mathbb C[X,Y,Z]/\langle X^2+Y^2-Z^2\rangle$?

Question

I am asked to find the coordinate ring of the variety $Z(X^2+Y^2-Z^2)$ in $\mathbb C^3$.So,the question reduces to finding the quotient ring $\mathbb C[X,Y,Z]/\langle X^2+Y^2-Z^2\rangle$.But I do not find resemblance with any known ring so that I can use first isomorphism theorem.So,what should be a proper way of approaching such questions?

Answer 1

The polynomials $$x=UV,\quad y=\frac{V^2-U^2}2,\quad z=\frac{U^2+V^2}2$$ generate a subring $$\Bbb C[x,y,z]\subset\Bbb C[U,V]$$ isomorphic to $\Bbb C[X,Y,Z]/(X^2+Y^2-Z^2).$