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