If O B is the semi-minor axis of an ellipse, F 1 and F 2 are its focii and the angle between F 1 B and F 2 B is a right angle, then the square of the eccentricity of the ellipse is

Step-by-step solution

In standard ellipse x 2 a 2 + y 2 b 2 = 1 Semi-major axis O A = a Semi-minor axis O B = b Focus F 1 ≡ - a e , 0 F 2 ≡ a e , 0 ∠ F 1 B F 2 = 9 0 ° O F 1 = O F 2 ⇒ ∠ F 1 B O = ∠ F 2 B O = 4 5 ° ⇒ a e b = tan 4 5 ° = 1 ⇒ b = a e Also, b 2 = a 2 1 - e 2 ⇒ a 2 e 2 = a 2 1 - e 2 ⇒ 2 e 2 = 1 ⇒ e 2 = 1 2