Let $k$ be an algebraically closed field of characteristic not equal to 2 and let $V \subset \mathbb{P}_{k}^{3}$ be a nonsingular quadric surface.

(a) Prove that $V$ is birational to $\mathbb{P}_{k}^{2}$.

(b) Prove that there exists a pair of disjoint lines on $V$.

(c) Prove that the affine variety $W=\mathbb{V}(x y z-1) \subset \mathbb{A}_{k}^{3}$ does not contain any lines.