the hyperbola tangent theorem

 The Hyperbola-Tangent Theorem If P is a hyperbola and 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 H0 : xy = 1. (1) The lines x = 0 and y = 0 do not meet H0. (2) For c ≠ 0, the lines x = c and y = c each meet H0 exactly once, (3) For m ≠ 0, the line y = mx+c, cuts H0 twice if m > 0 or c2+4m > 0, is a tangent to H0 if c2+4m = 0, does not meet H0 if c2+4m < 0. Proof By Theorem AC2, there is an affine transformation s mapping H to H0. As s is affine maps to the parallel family . 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 will be one other tangent parallel to M. The symmetry group of H contains a half-turn r about the centre C. Then r(M) is a tangent. Since r is a half-turn, r(M) is parallel to M - so is the other parallel in this direction. Clearly C lies on the line joining the points of contact.