1. Understanding the Ellipse and its Key Properties

To solve this problem, we need to recall the fundamental definitions and formulas associated with an ellipse. An ellipse is defined by its semi-major axis ($a$), semi-minor axis ($b$), and eccentricity ($e$). For an ellipse centered at the origin with its major axis along the x-axis, its standard equation is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Here, we assume $a > b > 0$. Even if the major axis were along the y-axis, the distances between foci and directrices, and the length of the latus rectum, would remain the same, just with $a$ and $