This problem is a classic example of connecting two conic sections through a common property – their foci. The core idea is to first determine the foci of the given ellipse, then use this information to find the unknown parameter in the hyperbola's equation, and finally calculate the length of the latus rectum for the hyperbola.

Essential Concepts and Formulas

Before we begin, let's recall the standard forms and key properties of ellipses and hyperbolas centered at the origin:

1. Ellipse (with major axis along the x-axis):
   • Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$.