Distance Between Foci of Hyperbola H from Ellipse E Properties

To find the distance between the foci of the hyperbola $\text{H}$, we first analyze the given properties of the ellipse $\text{E}$.

Step 1: Determine the parameters of Ellipse $\text{E}$

The standard equation of the ellipse centered at the origin is: $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1 \quad (A > B)$

Given:

• Eccentricity $e_{\text{E}} = \frac{\sqrt{3}}{2}$
• Directrices $x = \pm \frac{4\sqrt{6}}{3}$

The equation of the directrices of an ellipse is given by $x = \pm \frac{A}{e_{\text{E}}}$. Therefore: $\frac{A}{e_{\text{E}}} = \frac{4\sqrt{6}}{3}$

Substituting $e_{\text{E}} = \frac{\sqrt{3}}{2}$: $A = \frac{\sqrt{3}}{2} \cdot \frac{4\sqrt{6}}{3} = \frac{4\sqrt{18}}{6} = \frac{12\sqrt{2}}{6} = 2\sqrt{2}$

Thus, the length of the semi-major axis of $\text{E}$ is $A = 2\sqrt{2}$.

Next, we find the semi-minor axis $B$ using the relation $B^2 = A^2(1 - e_{\text{E}}^2)$: $B^2 = (2\sqrt{2})^2 \left(1 - \left(\frac{\sqrt{3}}{2}\right)^2\right) = 8 \left(1 - \frac{3}{4}\right) = 8 \cdot \frac{1}{4} = 2$ $B = \sqrt{2}$

Thus, the length of the minor axis of $\text{E}$ is: $2B = 2\sqrt{2}$

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Step 2: Determine the parameters of Hyperbola $\text{H}$

The standard equation of the hyperbola is: $\text{H} : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

From the problem statement:

1. The eccentricity of $\text{H}$ is equal to the length of the semi-major axis of $\text{E}$: $e_{\text{H}} = A = 2\sqrt{2}$

2. The length of the latus rectum of $\text{H}$ is equal to the length of the minor axis of $\text{E}$: $\frac{2b^2}{a} = 2B = 2\sqrt{2} \implies b^2 = \sqrt{2}a$

Using the relation between eccentricity and the axes of a hyperbola, $e_{\text{H}}^2 = 1 + \frac{b^2}{a^2}$: $(2\sqrt{2})^2 = 1 + \frac{\sqrt{2}a}{a^2}$ $8 = 1 + \frac{\sqrt{2}}{a}$ $7 = \frac{\sqrt{2}}{a} \implies a = \frac{\sqrt{2}}{7}$

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Step 3: Calculate the distance between the foci of Hyperbola $\text{H}$

The foci of hyperbola $\text{H}$ are located at $(\pm a e_{\text{H}}, 0)$. The distance between the foci is given by $2ae_{\text{H}}$: $\text{Distance} = 2ae_{\text{H}} = 2 \cdot \left(\frac{\sqrt{2}}{7}\right) \cdot (2\sqrt{2}) = \frac{2 \cdot 4}{7} = \frac{8}{7}$

Thus, the distance between the foci of the hyperbola $\text{H}$ is $\frac{8}{7}$.