Understanding the Ellipse: Key Concepts and Standard Forms

An ellipse is a beautiful conic section defined as the locus of all points in a plane such that the sum of the distances from two fixed points (called foci) is constant. For an ellipse centered at the origin $(0,0)$, its geometric properties and standard equation depend critically on the orientation of its major axis.

There are two primary standard forms for an ellipse centered at the origin:

1. Major axis along the x-axis: This occurs when the length of the semi-major axis ($a$) is greater than the length of the semi-minor axis ($b$), and the foci lie on the x-axis.

   • Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
   • Foci: $(\pm ae, 0)$
   • Directrices: $x = \pm \frac{a}{e}$
   • Relationship between $a, b,$ and eccentricity $e$: $b^2 = a^2(1-e^2)$

2. Major axis along the y-axis: This occurs when the length of the semi-major axis ($a$) is greater than the length of the semi-minor axis ($b$), and the foci lie on the y-axis.

   • Equation: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$
   • Foci: $(0, \pm ae)$
   • Directrices: $y = \pm \frac{a}{e}$
   • Relationship between $a, b,$ and eccentricity $e$: $b^2 = a^2(1-e^2)$ (Here, $a$ is always the semi-major axis, and $b$ is the semi-minor axis, so $b < a$).

The eccentricity, $e$, for an ellipse always satisfies $0 < e < 1$. It's a fundamental characteristic that measures how "stretched out" the ellipse is; an eccentricity closer to 0 means the ellipse is more circular, while an eccentricity closer to 1 means it's more elongated.

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Problem Statement: Decoding the Given Information

We are tasked with finding the equation of an ellipse based on the following clues:

• Its center is at the origin $(0,0)$. This simplifies our choice to one of the two standard forms above.
• Its eccentricity, $e = \frac{1}{2}$. This value will be used in key relationships.
• One of its directrices is $x = 4$. The form of this equation is crucial for determining the ellipse's orientation.

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Step-by-Step Solution

Step 1: Determine the Orientation of the Major Axis

• Concept: The form of the directrix equation (whether it's $x=k$ or