I want to show that:
Let $Q$ be a quadric surface of rank 3 in $\mathbf{P}^3$(over $\mathbf{C}$), then any line in $Q$ must pass through the only singular point of $Q$.
I know that up to a transform of coordinate, we can assume that $Q=\{x_0^2+x_1^2+x_2^2=0\}$ and therefore the singular point is $(0:0:0:1)$. But I have no idea how to proceed. Thanks for any help.
Suppose for contradiction that some line $l$ lies on $Q$ ($l\subset Q$) but does not go through the apex (=singular point) $S$ of $Q$.
Then the whole plane $\Pi$ determined by $S$ and $l$ is contained in $Q$ ($\Pi\subset Q$): indeed for any $L\in l\subset Q$ the line $\overline {SL}$ lies on $Q$ and $\Pi$ is the union of all those lines $\overline {SL}$ when $L$ runs through $l$.
But any quadric containing a plane (here $\Pi\subset Q$) has rank $\leq 2$ : contradiction to the hypothesis that $Q$ has rank $3$.