Question

If the latus rectum of an ellipse subtends an equilateral triangle at the centre of the ellipse ,then write the eccentricity of the ellipse

Answer 1

AB=AB'=BB'

ΔABB' is equilateral

AB=√((a-0)²+(0-b)²)

=√(a²+b²)

AB'=√((a-0)²+(0-(-b)²)

=√(a²+6)

BB'=2b

AB=BB' =>(√(a²+b²))² =(2b)²

=> a²+b²=4b²

=> a²=3b²

b²=a²(1-e²)

=> a²=3a²(1-e²)

=> a²=3a²(1-e²)

= 1/3=1-e²

=>e²=1-1/3=2/3

e=√2/3