Let λ x - 2 y = μ be a tangent to the hyperbola a 2 x 2 - y 2 = b 2 . Then λ a 2 - μ b 2 is equal to

Step-by-step solution

Given that λ x - 2 y = μ is a tangent to the curve a 2 x 2 - y 2 = b 2 So a 2 x 2 - λ x - μ 2 2 = b 2 ⇒ 4 a 2 - λ 2 x 2 + 2 λ μ x - μ 2 - 4 b 2 = 0 So for the line to be tangent, the condition is discriminant = 0 i.e. 4 λ 2 μ 2 + 4 4 a 2 - λ 2 μ 2 + 4 b 2 = 0 4 λ 2 b 2 - 4 a 2 μ 2 = 16 a 2 b 2 Hence, λ 2 a 2 - μ 2 b 2 = 4