1. Understanding the Fundamental Definition of an Ellipse

An ellipse is a conic section defined as the locus of a point $P$ such that its distance from a fixed point (called the focus, $F$) is a constant multiple, $e$ (called the eccentricity), of its perpendicular distance from a fixed line (called the directrix, $D$). Mathematically, this fundamental definition is expressed as: $PF = e \cdot PM$ where:

• $P(x,y)$ is any point on the ellipse.
• $F$ is the focus.
• $PM$ is the perpendicular distance from point $P$ to the directrix $D$.
• $e$ is the eccentricity. For an ellipse, $e$ always satisfies $0 < e < 1$.

Our goal is to find the length of the semi-major axis, denoted by $a$.

2. Identifying Given Parameters

From the problem statement, we are given:

• The focus $F$ is at the origin, so $F = (0,0)$.
• The directrix $D$ is the line $x=4$.
• The eccentricity $e = \frac{1}{2}$.

We will solve this problem using two methods: first, by directly applying the fundamental definition to derive the ellipse's equation, and second, by using a standard property relating the focus, directrix, and semi-major axis, which is more efficient.

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**Method 1: Deriving the Equation of the Ellipse