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

For hyperbola $\dfrac{{x}^{2}}{10} - \dfrac{{y}^{2}}{10\cos^{2}\theta} = 1$

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

For ellipse $\dfrac{{x}^{2}}{5\cos^{2}\theta} + \dfrac{{y}^{2}}{5} = 1$

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

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 $= \dfrac{2{b}^{2}}{a} = \dfrac{10\cos^{2}\theta}{\sqrt{5}} = \dfrac{20}{3\sqrt{5}} = \dfrac{4\sqrt{5}}{3}$