Let e 1 and e 2 be the eccentricities of the ellipse x 2 25 + y 2 b 2 = 1 b < 5 and the hyperbola x 2 16 - y 2 b 2 = 1 respectively satisfying e 1 e 2 = 1 . If α and β are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair α , β is equal to:

Step-by-step solution

e 1 = 1 - b 2 25 and e 2 = 1 + b 2 16 Given e 1 e 2 = 1 ⇒ e 1 e 2 2 = 1 ⇒ 1 - b 2 25 1 + b 2 16 = 1 ⇒ b 2 16 - b 2 25 - b 4 25 × 16 = 0 ⇒ 9 16 × 25 b 2 - b 4 25 × 16 = 0 ⇒ b 2 = 9 Thus, e 1 = 1 - 9 25 = 4 5 and e 2 = 1 + 9 16 = 5 4 Therefore, α = 2 5 e 1 = 8 and β = 2 4 e 2 = 10 ∴ α ,   β = 8 ,   10