Condition for a Line to be Tangent

Line y = mx + c is tangent to x²/a² − y²/b² = 1 iff c² = a²m² − b². Requires |m| > b/a. Note the sign difference from the ellipse (where c² = a²m² + b²).

$y = mx + c$ is tangent to $x^2/a^2 - y^2/b^2 = 1$ iff $c^2 = a^2m^2 - b^2$ and $|m| > b/a$.

Memory aid — sign pattern across conics:

The pattern for ellipse and hyperbola differs only in the sign of $b^2$, matching the sign difference in their equations.

Test: Is $y = 2x + 1$ tangent to $x^2/4 - y^2/3 = 1$?

$m = 2$, $a^2 = 4$, $b^2 = 3$, $c = 1$. Check: $c^2 = 1$, $a^2m^2-b^2 = 16-3 = 13 \neq 1$. Not tangent.

Is $y = 2x + \sqrt{13}$ tangent? $c^2 = 13 = a^2m^2-b^2$. Yes.