Chord of ellipse and line interactionHard

Area of Intersection of Two Ellipses

Let the ellipse, ${E_1}:\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ , a > b and ${E_2}:\frac{{{x^2}}}{{{A^2}}} + \frac{{{y^2}}}{{{B^2}}} = 1$ A < B have same eccentricity $\frac{1}{{\sqrt 3 }}$ Let the product of their lengths of latus rectums be $\frac{{32}}{{\sqrt 3 }}$ , and the distance between the foci of E₁ be 4. If E₁ and E₂ meet at A,B,C and D, then the area of the quadrilateral ABCD equals :

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