the hyperbola tangent theorem

The Hyperbola-Tangent 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 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.

By Theorem AC2, there is an affine transformation s mapping H to H0.
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
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.

affine tangents page