Conic Sections · Ellipse latus rectum and foci

Let each of the two ellipses

E_1 : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

(a > b)

and

E_2 : \dfrac{x^2}{A^2} + \dfrac{y^2}{B^2} = 1

(A < B)

have eccentricity

\dfrac45

. Let the lengths of the latus recta of

E_1

and

E_2

\ell_1

and

\ell_2

, respectively, such that

2\ell_1^2 = 9\ell_2

. If the distance between the foci of

E_1

is 8, then the distance between the foci of

E_2

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