Tangents to x²/a² − y²/b² = 1 with slope m, valid only when |m| > b/a (otherwise the line is parallel to or between the asymptotes and no tangent exists). Point of contact: (±a²m/√(a²m²−b²), ±b²/√(a²m²−b²)).
Substitute y = m x + c y = mx + c y = m x + c x 2 / a 2 − y 2 / b 2 = 1 x^2/a^2 - y^2/b^2 = 1 x 2 / a 2 − y 2 / b 2 = 1
x 2 a 2 − ( m x + c ) 2 b 2 = 1 \frac{x^2}{a^2} - \frac{(mx+c)^2}{b^2} = 1 a 2 x 2 − b 2 ( m x + c ) 2 = 1 b 2 x 2 − a 2 ( m x + c ) 2 = a 2 b 2 b^2x^2 - a^2(mx+c)^2 = a^2b^2 b 2 x 2 − a 2 ( m x + c ) 2 = a 2 b 2 ( b 2 − a 2 m 2 ) x 2 − 2 a 2 m c x − a 2 c 2 − a 2 b 2 = 0 (b^2 - a^2m^2)x^2 - 2a^2mcx - a^2c^2 - a^2b^2 = 0 ( b 2 − a 2 m 2 ) x 2 − 2 a 2 m c x − a 2 c 2 − a 2 b 2 = 0 For tangency, discriminant = 0 = 0 = 0 b 2 − a 2 m 2 ≠ 0 b^2 - a^2m^2 \neq 0 b 2 − a 2 m 2 = 0
( 2 a 2 m c ) 2 + 4 ( b 2 − a 2 m 2 ) ( a 2 c 2 + a 2 b 2 ) = 0 (2a^2mc)^2 + 4(b^2-a^2m^2)(a^2c^2+a^2b^2) = 0 ( 2 a 2 m c ) 2 + 4 ( b 2 − a 2 m 2 ) ( a 2 c 2 + a 2 b 2 ) = 0 4 a 4 m 2 c 2 + 4 a 2 ( b 2 − a 2 m 2 ) ( c 2 + b 2 ) = 0 4a^4m^2c^2 + 4a^2(b^2-a^2m^2)(c^2+b^2) = 0 4 a 4 m 2 c 2 + 4 a 2 ( b 2 − a 2 m 2 ) ( c 2 + b 2 ) = 0 a 2 m 2 c 2 + b 2 c 2 − a 2 m 2 c 2 + b 4 − a 2 m 2 b 2 = 0 a^2m^2c^2 + b^2c^2 - a^2m^2c^2 + b^4 - a^2m^2b^2 = 0 a 2 m 2 c 2 + b 2 c 2 − a 2 m 2 c 2 + b 4 − a 2 m 2 b 2 = 0 b 2 c 2 + b 4 − a 2 m 2 b 2 = 0 ⟹ c 2 = a 2 m 2 − b 2 b^2c^2 + b^4 - a^2m^2b^2 = 0 \implies c^2 = a^2m^2 - b^2 b 2 c 2 + b 4 − a 2 m 2 b 2 = 0 ⟹ c 2 = a 2 m 2 − b 2 For real c c c c 2 > 0 ⇒ a 2 m 2 > b 2 ⇒ ∣ m ∣ > b / a c^2 > 0 \Rightarrow a^2m^2 > b^2 \Rightarrow |m| > b/a c 2 > 0 ⇒ a 2 m 2 > b 2 ⇒ ∣ m ∣ > b / a
The tangents: y = m x ± a 2 m 2 − b 2 y = mx \pm \sqrt{a^2m^2 - b^2} y = m x ± a 2 m 2 − b 2
When ∣ m ∣ = b / a |m| = b/a ∣ m ∣ = b / a c = 0 c = 0 c = 0 y = m x y = mx y = m x
When ∣ m ∣ < b / a |m| < b/a ∣ m ∣ < b / a c 2 < 0 c^2 < 0 c 2 < 0