= Solution
Over an algebraically closed field of characteristic different from two, a projective change of coordinates puts every <smooth quadric surface> in the form
$$
V(x_0x_3-x_1x_2).
$$
The <Segre embedding> identifies this surface with $\mathbb P^1\times\mathbb P^1$ by
$$
([s:t],[u:v])\longmapsto[su:sv:tu:tv].
$$
Two distinct fibres of either coordinate projection are disjoint lines in one of the <rulings of a smooth quadric surface>. Explicitly,
$$
L_1=\{x_2=x_3=0\},\qquad
L_2=\{x_0=x_1=0\}
$$
both lie on $V$ and have no common projective point. Thus $V$ contains a pair of disjoint lines.
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