An ellipse has O B as semi-minor axis, S and S^ are foci and angle S B S^ is a right angle. Then the eccentricity of the ellipse is

Step-by-step solution

Given an ellipse defined by x^2 a^2 + y^2 b^2 = 1 with foci S(ae, 0) and S'(-ae, 0) , and point B(0, b) on the minor axis. The condition SBS' = 90^ implies orthogonality of vectors BS = (ae, -b) and BS' = (-ae, -b) . Their dot product yields -a^2e^2 + b^2 = 0 , giving b^2 = a^2e^2 . Using the standard identity b^2 = a^2(1 - e^2) and substituting, we obtain: a^2e^2 = a^2(1 - e^2) Dividing by a^2 yields e^2 = 1 - e^2 , hence 2e^2 = 1 and e^2 = 1 2 . Since 0 Final answer: 1 2