For 0 < θ < π / 2 , if the eccentricity of the hyperbola x 2 − y 2 cosec 2 θ = 5 is 7 times eccentricity of the ellipse x 2 cosec 2 θ + y 2 = 5 , then the value of θ is:

Step-by-step solution

Given: The equation of hyperbola is, x 2 - y 2 cosec 2 θ = 5 ⇒ x 2 5 - y 2 5 sin 2 θ = 1 And the equation of ellipse is x 2 cosec 2 θ + y 2 = 5 ⇒ x 2 5 sin 2 θ + y 2 5 = 1 here a < b as sin 2 θ ≤ 1 So, e H = 1 + sin 2 θ & e E = 1 - sin 2 θ Also given, e H = 7 e E ⇒ 1 + sin 2 θ = 7 1 - sin 2 θ ⇒ 1 + sin 2 θ = 7 - 7 sin 2 θ ⇒ 8 sin 2 θ = 6 ⇒ sin θ = 3 2 ⇒ θ = π 3