This problem combines geometric properties of an ellipse with its algebraic definition. We are given information about a right-angled triangle formed by the foci and an extremity of the minor axis, and its area. Our goal is to find the length of the latus rectum.

1. Setting Up the Ellipse and Key Coordinates

Let the standard equation of the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ is the semi-major axis length and $b$ is the semi-minor axis length. The relationship between $a$, $b$, and the eccentricity $e$ is given by $b^2 = a^2(1-e^2)$.

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