Animated Solution for Mathematics - Conic Sections: A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $\frac{1}{2}$. Then the length of the semi-major axis is

We begin with a focus $S$ located at the origin $(0,0)$ and a directrix defined by the line $x=4$.

The distance between the focus and the directrix is the physical gap we must bridge. Since the focus is at $x=0$ and the directrix is at $x=4$, the distance is simply $4-0=4$.

We connect this physical reality to the abstract properties of an ellipse. For any ellipse, the distance from the center to the focus is $ae$, and the distance from the center to the directrix is $\frac{a}{e}$.

Because the focus and the directrix lie on the same side of the center, the distance between them is given by the difference:

This equation serves as the fundamental key to unlocking the parameters of our conic section.

We now substitute the given eccentricity $e=\frac{1}{2}$ into our master equation. First, we factor out $a$:

Substituting $e=\frac{1}{2}$ into the expression, we obtain:

Simplifying the reciprocal inside the parentheses yields:

This further simplifies to:

To isolate the semi-major axis $a$, we multiply both sides of the equation by $\frac{2}{3}$:

The length of the semi-major axis is $\frac{8}{3}$. This elegant result demonstrates how coordinate geometry translates spatial relationships into precise algebraic truths.