Condition for a Line to be Tangent

Line y = mx + c is tangent to x²/a² + y²/b² = 1 iff c² = a²m² + b². Unlike a parabola, two values of c (opposite signs) work for each slope m.

From the tangent-slope derivation: $y = mx + c$ is tangent to $x^2/a^2 + y^2/b^2 = 1$ iff:

$$c^2 = a^2m^2 + b^2$$

Comparison across conics:

The parabola is the only conic with a unique tangent per slope (its one "open end" allows only one tangent direction per slope).

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

Here $m = 1$, $a^2 = 4$, $b^2 = 3$, $c = 3$.

Check: $c^2 = 9$, $a^2m^2 + b^2 = 4 + 3 = 7 \neq 9$. Not a tangent.

Is $y = x + \sqrt{7}$ tangent? $c^2 = 7 = a^2m^2+b^2 = 7$. Yes.