The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \(\frac{\sqrt{3}}{2}\) and its directrices are \…

The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \(\frac{\sqrt{3}}{2}\) and its directrices are \(x= \pm \frac{4 \sqrt{6}}{3}\). Let \(H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \(E\) , and whose length of latus rectum is equal to the length of minor axis of \(E\). Then the distance between the foci of \(H\) is:

[JEE Main 2026, 6 Apr (Shift 2)]