Parametrization of a conic by projection

For a nonsingular conic with a point $p$ and a projective line $L$ not containing $p$, send $q\in L$ to the second intersection of $pq$ with the conic, counted with multiplicity. The tangent direction gives $p$ itself. The inverse sends any other conic point to its joining line's intersection with $L$, and sends $p$ to the tangent's intersection. The Bézout theorem makes these maps inverse. With data over a field, the construction preserves rationality: a known rational root of the restricted quadratic equation leaves its other root rational too. A conic with a rational point is therefore rationally parametrized by a projective line.