If the eccentricity of the hyperbola x 2 1 + sin ⁡ θ 2 - y 2 cos 2 ⁡ θ = 1 is 2 3 , then the sum of all the possible values of θ is (where, θ ∈ 0 , π )

Step-by-step solution

e 2 = 1 + cos 2 ⁡ θ 1 + sin ⁡ θ 2 = 4 3 ⇒ cos 2 ⁡ θ 1 + sin ⁡ θ 2 = 1 3 ⇒ cos ⁡ θ 1 + sin ⁡ θ 2 = 1 3 ⇒ sin ⁡ π 2 - θ 1 + cos ⁡ π 2 - θ 2 = 1 3 ⇒ tan 2 ⁡ π 4 - θ 2 = 1 3 π 4 - θ 2 = - π 6 , π 6 ⇒ θ 2 = π 4 - π 6 , π 4 + π 6 θ = π 2 ± π 3 = π 6 , 5 π 6 Hence, the required sum = π