If S and S ' are the focii of an ellipse, B is one end of the minor axis and ∠ S B S ' = 90 ° , then the eccentricity of that ellipse is

Step-by-step solution

Let the coordinates of the focii of an ellipses are S a e , 0 and S ' - a e , 0 . Since, ∠ S B S ' = 90 ∴ S S ' 2 = S B 2 + S ' B 2 2 a e 2 = a e - 0 2 + 0 - b 2 2 + - a e - 0 2 + 0 - b 2 2 ⇒ 2 a e 2 = a e 2 + b 2 + a e 2 + b 2 2 ⇒ 4 a 2 e 2 = 2 a 2 e 2 + 2 b 2 ⇒ 2 a 2 e 2 = 2 b 2 ⇒ b 2 a 2 = e 2     . . . i Also, e 2 = 1 - b 2 a 2 ⇒ e 2 = 1 - e 2 ;   Using   i ⇒ e = 1 2 .