klein view page

tangents to a hyperbola

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 c²+4m > 0,
is a tangent to H if c²+4m = 0,
does not meet H if c²+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. mx²+cx-1 = 0.
This quadratc has discriminant c²+4m, so that

there are two roots if c²+4m > 0, and, in particular, if m > 0.
Thus the line cuts the hyperbola twice in these cases.
there are two coincident roots if c²+4m = 0, so the line is a tangent to H,
there are no real roots if c²+4m < 0, so the line does not meet H.

special conics page

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 c²+4m > 0, is a tangent to H if c²+4m = 0, does not meet H if c²+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. mx²+cx-1 = 0. This quadratc has discriminant c²+4m, so that there are two roots if c²+4m > 0, and, in particular, if m > 0. Thus the line cuts the hyperbola twice in these cases. there are two coincident roots if c²+4m = 0, so the line is a tangent to H, there are no real roots if c²+4m < 0, so the line does not meet H.