Ellipses with Same Eccentricity

Let E1:

x2

a2

+

y2

b2

=1,a>b. Let E2 be another ellipse such that it touches the end points of major axis of E1 and the foci ofE2 are the end points of minor axis of E1. If E1 and E2 have same eccentricities, then its value is: