The ParabolaTangent Theorem If P is a parabola and <L> is a family of lines, then (1) with one exception, each family contains exactly one tangent to P, and (2) in the exceptional case, each member of the family cuts C once.
This depends on the result for a standard parabola: The Parabola P_{0} : y^{2} = x.
(1) The line y = k cuts P_{0} exactly once.


Proof By Theorem AC2, there is an affine transformation s mapping P to P_{0}. As s is affine <L> maps to the parallel family <s(L)>. By the Parabola Theorem, the latter family contains exactly one tangent M. As s^{1} is also affine, s^{1}(M) is the only tangent in <L>.

