Suppose that $ k $ is a field of characteristic $ \neq 2. $ Decompose into irreducible components the closed set $ X \subset \mathbb{A}^{3} $ defined by $ x^{2}+y^{2}+z^{2} = 0, x^{2}-y^{2}-z^{2}+1 = 0. $
My impression is that $ V(x^{2}+y^{2}+z^{2}) \cap V(x^{2}-y^{2}-z^{2}+1) = \emptyset. $
$ \emptyset $ is clearly irreducible, but I feel that I have gone horribly wrong.