Is the set of solution of $x_0^2+x_1^2-x_2^2=0$ homeomorphic to the set of solution of $x_0^2-x_1^2=0$ in $\mathbb{P}^2(\mathbb{R})$?

Question

Is the set of solution to $x_0^2+x_1^2-x_2^2=0$ homeomorphic to the set of solution to $x_0^2-x_1^2=0$ in $ \mathbb{P}^2 (\mathbb{R})$? I think one can show that the first set is homeomorphic to $S^1$, and that by removing the point $[0,0,1]$ from the second it is no more connected, hence they aren't homeomorphic. However I would like to learn other solutions or have a confirmation.