Let 0 < θ < π 2 . If the eccentricity of the hyperbola x 2 cos 2 ⁡ θ - y 2 sin 2 ⁡ θ = 1 is greater than 2 , then the length of its latus rectum lies in the interval:

Step-by-step solution

For given hyperbola the eccentricity is given as e 2 = 1 + s i n 2 θ c o s 2 θ = 1 + t a n 2 θ = s e c 2 θ ⇒ e = s e c θ ∴ Length of latus rectum l = 2 s i n 2 θ c o s θ = 2 t a n 2 θ s e c θ ⇒ l = 2 ( e 2 - 1 ) e = 2 e - 1 e On differentiating w.r.t. e , we get d l d e = 2 1 + 1 e 2 > 0 ∴ l is an increasing function ∴ l m i n = 2 2 - 1 2 = 3 ∴   Range of latus rectum is 3 , ∞ .