The HyperbolaTangent Theorem If P is a hyperbola and <L> is a family of lines, then there are three cases (1) the family contains exactly two tangents to P, or (2) each member of the family cuts H twice, or (3) one member of the family does not meet H, all others cut H once. (4) There are exactly two families of the third type. (5) The tangents to H at P and Q are parallel if and only if PQ is a diameter.
This depends on the result for a standard hyperbola: The Hyperbola H_{0} : xy = 1.
(1) The lines x = 0 and y = 0 do not meet H_{0}.


Proof By Theorem AC2, there is an affine transformation s mapping H to H_{0}. As s is affine <L> maps to the parallel family <s(L)>. By the Hyperbola Theorem, the results (1)  (4) follow at once.
For the last part, we note that if M is a tangent, then we know that there

