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^ ?

Step-by-step solution

Let ellipse be x^2 a^2 + y^2 b^2 =1 It's foci are S ₁(a e, o) and S ₂(-a e, o) and the extremity of the minor axis is B (o, b) Then, slope of BS ₂= b-o o-(-a e) = b a e and slope of BS ₁= b-o o-a e =- b a e The product of slopes = b a e - b a e =-1 b^2=a^2 e^2 e^2= b^2 a^2 ....(i) We have, e= c a c= a^2-b^2 e^2=1- b^2 a^2 =1-e^2 [ from (i)] aligned & & 2 e^2 & =1 & & e^2 & = 1 2 e= 1 2 aligned