Mathematics

The key is to use the focus-directrix definition of an ellipse: any point on the ellipse is at a distance from the focus that equals the eccentricity times its distance to the directrix. Set up this relationship for a general point, then use it to find the semi-major axis. Since you know the focus location, directrix, and eccentricity $e = \frac{1}{2}$, you can determine how the ellipse is positioned and extract $a$ from the geometric constraints.

The key is to use the focus-directrix definition of an ellipse: any point on the ellipse is at a distance from the focus that equals the eccentricity times its distance to the directrix. Set up this relationship for a general point, then use it to find the semi-major axis. Since you know the focus location, directrix, and eccentricity $e = \frac{1}{2}$, you can determine how the ellipse is positioned and extract $a$ from the geometric constraints.