1. Fundamental Concepts and Formulas for an Ellipse

To tackle this problem effectively, we must first recall the standard definitions and relationships associated with an ellipse. We typically consider an ellipse centered at the origin $(0,0)$ with its major axis lying along the x-axis.

• Standard Equation of an Ellipse: The canonical equation for such an ellipse is given by: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Here, $a$ represents the length of the semi-major axis (half the length of the major axis), and $b$ represents the length of the semi-minor axis (half the length of the minor axis). For this orientation, it's a fundamental assumption that $a > b > 0$.

• Length of the Minor Axis: The total length of the minor axis is $2b$.

• Foci: The foci are two crucial fixed points on the major axis, equidistant from the center. Their coordinates are $(\pm ae, 0)$, where $e$ is the eccentricity of the ellipse.

• Distance Between the Foci: The distance separating the two foci $(\pm ae, 0)$ is calculated as $ae - (-ae) = 2ae$.

• Eccentricity ($e$): Eccentricity is a dimensionless quantity that describes how "stretched out" or "oval" an ellipse is. For an ellipse, its value always lies between 0 and 1 ($0 < e < 1$). A value closer to 0 indicates a shape closer to a circle, while a value closer to 1 indicates a more elongated shape. The eccentricity is intrinsically linked to $a$ and $b$ by the fundamental identity: $b^2 = a^2(1 - e^2)$ This identity is paramount for solving many ellipse problems as it connects the dimensions of the ellipse ($a$, $b$) with its shape parameter ($e$). It can also be rearranged as $e^2 = 1 - \frac{b^2}{a^2}$.

  Tip: Always remember the fundamental identity $b^2 = a^2(1 - e^2)$. It's the most common bridge between the semi-axes and eccentricity.

2. Translating the Given Condition into a Mathematical Equation

The problem statement provides a specific condition: "the length of the minor axis of an ellipse is equal to half of the distance between the foci." Our first step is to translate this verbal statement into a precise mathematical equation using the definitions from Section 1.

• Length of the minor axis: From our definitions, this is $2b$.
• Distance between the foci: From our definitions, this is $2ae$.
• "equal to half of": This means we multiply the distance between foci by $\frac{1}{2}$.

Combining these, the given condition translates to: $2b = \frac{1}{2} (2ae)$ This equation now establishes a direct relationship between the semi-minor axis ($b$), the semi-major axis ($a$), and the eccentricity ($e$) based on the problem's specific premise.

3. Step-by-Step Derivation of the Eccentricity

Our ultimate goal is to determine the value of the eccentricity, $e$. We now have two crucial equations at our disposal:

1. The condition derived from the problem statement: $2b = \frac{1}{2}(2ae)$
2. The fundamental relationship for an ellipse: $b^2 = a^2(1 - e^2)$

Let's proceed with the derivation:

• Step 1: Simplify the given condition and express $b$ in terms of $a$ and $e$. We begin by simplifying the equation obtained from the problem statement: $2b = \frac{1}{2} (2ae)$ $2b = ae$ Now, to prepare for substitution into the fundamental identity (which involves $b^2$), let's isolate $b$: $b = \frac{ae}{2}$ Why this step? By simplifying the initial condition and expressing $b$ explicitly, we create a direct link between $b$, $a$, and $e$. This form is convenient for substitution, as it allows us to eliminate $b$ from our equations.

• Step 2: Utilize the fundamental identity of an ellipse. Recall the fundamental identity that relates the semi-major axis, semi-minor axis, and eccentricity: $b^2 = a^2(1 - e^2)$ Why this step? This identity is the core property of an ellipse that connects its dimensions to its eccentricity. We need to use this to solve for $e$.

• Step 3: Substitute the expression for $b$ into the fundamental identity. Now, substitute the expression for $b$ from Step 1 ($b = \frac{ae}{2}$) into the fundamental identity from Step 2: $\left( \frac{ae}{2} \right)^2 = a^2(1 - e^2)$ Why this step? This is the crucial step where we combine the specific information given in the problem with the general properties of an ellipse. By substituting, we eliminate $b$ from the equation, resulting in an equation that contains only $a$ and $e$. This allows us to isolate and solve for $e$.

• Step 4: Simplify the equation and solve for $e$. First, expand the left side of the equation: $\frac{a^2e^2}{4} = a^2(1 - e^2)$ Since $a$ represents the length of the semi-major axis, it must be a non-zero value ($a \neq 0$) for a valid ellipse. Therefore, we can safely divide both sides of the equation by $a^2$: $\frac{e^2}{4} = 1 - e^2$ Now, we rearrange the terms to collect all terms involving $e^2$ on one side: $e^2 = 4(1 - e^2)$ $e^2 = 4 - 4e^2$ Add $4e^2$ to both sides: $e^2 + 4e^2 = 4$ $5e^2 = 4$ Divide by 5: $e^2 = \frac{4}{5}$ Finally, take the square root of both sides to find $e$. Since eccentricity $e$ is a positive physical quantity, we take the positive square root: $e = \sqrt{\frac{4}{5}}$ $e = \frac{\sqrt{4}}{\sqrt{5}}$ $e = \frac{2}{\sqrt{5}}$ Why these steps? These are standard algebraic manipulations aimed at isolating the variable $e$. Dividing by $a^2$ is valid because $a$ is a non-zero length. Collecting like terms and then solving for $e^2$ helps simplify the equation. Taking the positive square root ensures that $e$ adheres to its definition as a positive value.

• Step 5: Rationalize the denominator (optional, but good practice). To present the answer in a standard form, we can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{5}$: $e = \frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$

4. Comparing with Options and Addressing Discrepancy

Let's review our calculated eccentricity and compare it with the provided options: Our calculated value is $e = \frac{2}{\sqrt{5}}$.

The given options are: (A) $\frac{1}{\sqrt{3}}$ (B) $\frac{2}{\sqrt{5}}$ (C) $\frac{\sqrt{3}}{2}$ (D) $\frac{\sqrt{5}}{3}$

Our derived eccentricity $e = \frac{2}{\sqrt{5}}$ perfectly matches Option (B).

Important Note on Discrepancy: The problem statement indicates that the "Correct Answer" is (A), which is $\frac{1}{\sqrt{3}}$. However, based on the standard definitions of an ellipse and the condition given in the problem ("length of the minor axis of an ellipse is equal to half of the distance between the foci"), the eccentricity is unequivocally derived as $e = \frac{2}{\sqrt{5}}$.

If the eccentricity were indeed $\frac{1}{\sqrt{3}}$, the initial condition provided in the problem statement would have to be different. For example, let's verify: If $e = \frac{1}{\sqrt{3}}$, then $e^2 = \frac{1}{3}$. Using the fundamental identity: $b^2 = a^2(1 - e^2) = a^2(1 - \frac{1}{3}) = a^2\left(\frac{2}{3}\right)$. So, $b = a\sqrt{\frac{2}{3}}$. Now, let's check if the original condition $2b = ae$ holds: $2\left(a\sqrt{\frac{2}{3}}\right) = a\left(\frac{1}{\sqrt{3}}\right)$ $2\sqrt{2} \cdot \frac{a}{\sqrt{3}} = \frac{a}{\sqrt{3}}$ $2\sqrt{2} = 1$ This is clearly false. Therefore, the problem statement as given leads to $e = \frac{2}{\sqrt{5}}$, not $e = \frac{1}{\sqrt{3}}$. There appears to be an inconsistency between the problem statement/solution and the 'Correct Answer' provided in the prompt. We have followed the problem statement rigorously.

5. Summary and Key Takeaways

This problem is a classic example of how to use the fundamental properties of an ellipse to derive an unknown parameter.

• Key Skill 1: Translation: Accurately translating the verbal description of geometric properties into mathematical equations is the crucial first step.
• Key Skill 2: Fundamental Identity: The relationship $b^2 = a^2(1 - e^2)$ is indispensable for solving problems involving the eccentricity of an ellipse. Always have this formula ready.
• Key Skill 3: Algebraic Manipulation: Be proficient in simplifying equations, substituting expressions, and solving for the desired variable. Remember that $a \neq 0$ for a non-degenerate ellipse, and $e$ must be positive.

By systematically applying these concepts, we derived the eccentricity $e = \frac{2