For some θ ∈ (0, (π)/(2)), if the eccentricity of the hyperbola, x^...

For hyperbola& $\dfrac{x2}{10} - \dfrac{y2}{10\cos^{2}\theta} = 1$

The eccentricity of the hyperbola~${e}_{H} = \sqrt{1 + \dfrac{b2}{a2}} = \sqrt{1 + \cos^{2}\theta}$

For ellipse~$\dfrac{x2}{5\cos^{2}\theta} + \dfrac{y2}{5} = 1$

The eccentricity of an ellipse~${e}_{E} = \sqrt{1 - \dfrac{b2}{a2}} = \sqrt{1 - \cos^{2}\theta} = sin\theta$

Given,~${e}_{H} = \sqrt{5}{e}_{E}$

$⇒ 1 + \cos^{2}\theta = 5\sin^{2}\theta ⇒ \cos^{2}\theta = \dfrac{2}{3}$

Now,~the length of the latus rectum~of an ellipse~$= \dfrac{2b^{2}}{a} = \dfrac{10\cos^{2}\theta}{5} = \dfrac{20}{3\sqrt{5}} = \dfrac{4\sqrt{5}}{3}$