What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the minor axis is 90°?

Let the ellipse be x²/a² + y²/b² = 1. Foci are (±ae, 0) and extremity of minor axis is (0, b). The lines from (ae, 0) and (–ae, 0) to (0, b) are perpendicular when the triangle is right-angled at (0, b). By geometry/Pythagoras: a² = (ae)² + (ae)² (since distance from each focus to (0,b) equals a, the semi-major axis), so a² = 2a²e², giving e² = 1/2, e = 1/√2.