It seems that there is a standard technique for an idea similar to the stereographic projection. I don't know how can I use it. For example here in this exercise, how can I use it ? I'm really sorry for being so stupid....
Define a birational map from an irreducible quadric hypersurface $X \subset P^3$ to $P^2$, by analogy with the stereographic projection, and find the open sets $U \subset X$, $V \subset P^2$, that are isomorphic.