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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.

Proof
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

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.
Proof 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.