If the eccentricity of the hyperbola x² – y² sec²∝ = 5 is times the eccentricity of the ellipse x² sec²∝ +y² = 25, then ∝ =

Given: e₁ and e₂ are respectively the eccentricities of x² – y² sec² α = 5 and x² sec² α + y² = 25 respectively

To find: value of α

x² – y² sec² α = 5

Eccentricity(e) of hyperbola is given by,

Therefore,

For ellipse:

x² sec² α + y² = 25

Eccentricity(e) of ellipse is given by,

Therefore,

According to question:

Eccentricity of given hyperbola is time eccentricity of given ellipse

From (1) and (2):

Squaring both sides:

⇒ 1 + cos² α = 3(1 – cos² α)

⇒ 1 + cos² α = 3 – 3 cos² α

⇒ 3 cos² α + cos² α = 3 – 1

⇒ 4 cos² α = 2