Tangents are drawn from the point α ,   β to the hyperbola 3 x 2 - 2 y 2 = 6 and are inclined at angles θ and ϕ to the x -axis. If tan ⁡ θ tan ⁡ ϕ = 2 , then the value of 2 α 2 - β 2 is

Step-by-step solution

Given hyperbola is 3 x 2 - 2 y 2 = 6 or x 2 2 - y 2 3 = 1 Slope form of tangent is y = m x ± a 2 m 2 - b 2 or y - m x 2 = a 2 m 2 - b 2 Tangent from the point α ,   β is given by, β - m α 2 = 2 m 2 - 3 ⇒ m 2 α 2 - 2 - 2 α β m + β 2 + 3 = 0 m 1 m 2 = β 2 + 3 α 2 - 2 = 2 = tan ⁡ θ tan ⁡ ϕ ∴ β 2 + 3 = 2 α 2 - 4 ⇒ 7 = 2 α 2 - β 2