The Hyperbola H : xy = 1.
(1) The lines x = 0 and y = 0 do not meet H.
(2) For c ≠ 0, the lines x = c and y = c each meet H exactly once,
(3) For m ≠ 0, the line y = mx+c,
 cuts H twice if m > 0 or c^{2}+4m > 0,
 is a tangent to H if c^{2}+4m = 0,
 does not meet H if c^{2}+4m < 0.


Proof
(1) If x or y = 0, then we cannot have xy = 1, so the line does not meet H.
(2) If x = c ≠ 0, then the line meets H at (c,1/c). Likewise, the line y = c ≠ 0
meets H at (1/c,c).
(3) The line y = mx+c meets H where x(mx+c) = 1, i.e. mx^{2}+cx1 = 0.
This quadratc has discriminant c^{2}+4m, so that
 there are two roots if c^{2}+4m > 0, and, in particular, if m > 0.
Thus the line cuts the hyperbola twice in these cases.
 there are two coincident roots if c^{2}+4m = 0, so the line is a tangent to H,
 there are no real roots if c^{2}+4m < 0, so the line does not meet H.

