Understanding the Core Relationships of an Ellipse

To solve this problem efficiently, we must first recall the fundamental definitions and the key algebraic relationship governing an ellipse. For an ellipse centered at the origin, with its major axis along the x-axis, its standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ is the semi-major axis and $b$ is the semi-minor axis, with $a > b > 0$.

The most crucial relationship connecting the semi-major axis ($a$), semi-minor axis ($b$), and eccentricity ($e$) is: $b^2 = a^2(1 - e^2)$ This equation can be rearranged into a very useful form, especially when dealing with the foci: $b^2 = a^2 - a^2e^2$ Since the foci of the ellipse are located at $(\pm ae, 0)$, the distance from the center to a focus is $ae$. Thus, we can write: $b^2 = a^2 - (ae)^2$ This form is particularly powerful because it directly relates the semi-minor axis ($b$) to the semi-major axis ($a$) and the focal distance ($ae$).

Problem Breakdown: Identifying Given Information and the Goal

We are provided with two pieces of information about an ellipse:

1. The distance between its foci is $6$.
2. The length of its minor axis is $8$.

Our objective is to calculate the eccentricity ($e$) of this ellipse.

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Step-by-Step Solution

Step 1: Translate Given Information into Ellipse Parameters

The first crucial step is to convert the verbal descriptions into mathematical expressions using the standard notation for an ellipse.

• Given: "The distance between its foci is $6$."

  • Why this is important: For any ellipse, the two foci are located at a distance of $ae$ from the center along the major axis. Therefore, the total distance between the two foci is $ae - (-ae) = 2ae$.
  • Calculation: $2ae = 6$

• Given: "The length of its minor axis is $8$."

  • Why this is important: The minor axis of an ellipse has a total length of $2b$, where $b$ is the semi-minor axis.
  • Calculation: $2b = 8$

Step 2: Determine the Values of the Focal Distance ($ae$) and Semi-Minor Axis ($b$)

Now, we simplify the equations from Step 1 to find the individual values of $ae$ (the distance from the center to a focus) and $b$ (the semi-minor axis). These values are direct components needed for the fundamental relationship.

• From the distance between foci: $2ae = 6 \implies ae = \frac{6}{2} \implies ae = 3$

  • Why: This gives us the value of $ae$, which is a key component in both the fundamental relationship $b^2 = a^2 - (ae)^2$ and for calculating eccentricity ($e = \frac{ae}{a}$).

• From the length of the minor axis: $2b = 8 \implies b = \frac{8}{2} \implies b = 4$

  • Why: This gives us the value of $b$, which is directly used in the fundamental relationship $b^2 = a^2 - (ae)^2$.

Step 3: Utilize the Fundamental Relationship to Find the Semi-Major Axis ($a$)

We will now employ the fundamental relationship $b^2 = a^2 - (ae)^2$ because we have already determined the values for $b$ and $ae$.

• Why this form is chosen: This specific form is incredibly efficient here because we have direct values for $b$ and $ae$. Substituting these values allows us to directly solve for $a^2$, and subsequently $a$, which is necessary to find the eccentricity.
• Calculations: Substitute $b=4$ and $ae=3$ into the equation $b^2 = a^2 - (ae)^2$: $(4)^2 = a^2 - (3)^2$ $16 = a^2 - 9$ Now, solve for $a^2$: $a^2 = 16 + 9$ $a^2 = 25$ Since $a$ represents a length (the semi-major axis), it must be positive. Taking the square root: $a = \sqrt{25} \implies a = 5$
  • Why finding $a$ is crucial: We need $a$ because eccentricity is defined as $e = \frac{ae}{a}$. Having $ae$ and $a$ allows for direct calculation of $e$.

Step 4: Calculate the Eccentricity ($e$)

With the values of $ae$ and $a$ now determined, we can directly calculate the eccentricity.

• Why: The definition of eccentricity for an ellipse is $e = \frac{\text{distance from center to focus}}{\text{semi-major axis}} = \frac{ae}{a}$. We have both components.
• Calculations: We found $ae = 3$ and $a = 5$. $e = \frac{ae}{a}$ Substitute the values: $e = \frac{3}{5}$

Step 5: Verify the Eccentricity Value

• Why this check is important: For any ellipse, the eccentricity $e$ must always satisfy the condition $0 < e < 1$. This verification step helps confirm the mathematical plausibility of our answer and can catch potential calculation errors.
• Verification: Our calculated eccentricity $e = \frac{3}{5} = 0.6$. This value indeed falls within the valid range for an ellipse ($0 < 0.6 < 1$), confirming our result. Also, we observe that $a=5$ and $b=4$, so $a>b$, which is consistent with the definition of $a$ as the semi-major axis.

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Final Answer

The eccentricity of the ellipse is $\boxed{\text{(A)} {3 \over 5}}$.

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JEE Main Tips & Common Pitfalls to Avoid

1. Memorize Key Definitions and Formulas: A strong grasp of definitions like major axis ($2a$), minor axis ($2b$), distance between foci ($2ae$), and the fundamental relationship $b^2 = a^2(1 - e^2)$ is non-negotiable for conic sections.
2. Master the Rearranged Fundamental Equation: The form $b^2 = a^2 - (ae)^2$ is incredibly powerful. Recognize when to use it (e.g., when $b$ and $ae$ are known or easily derivable) as it often provides the most direct path to finding $a$ or $e$.
3. Positive Lengths: Remember that $a$, $b$, and $ae$ represent physical lengths or distances, so their values must always be positive. If you get a negative value, recheck your algebra.
4. Eccentricity Range Check: Always perform a quick check to ensure your calculated eccentricity $e$ for an ellipse lies between $0$ and $1$ (i.e., $0 < e < 1$). If it doesn't, there's likely an error in your calculation.
5. Distinguish $a$ and $b$: Always remember that $a$ is the semi-major axis and $b$ is the semi-minor axis. For an ellipse, $a$ is always greater than $b$ ($a > b$). This distinction is crucial.

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Summary and Key Takeaway

This problem is a classic application of the fundamental properties of an ellipse. The key to solving it efficiently was to:

1. Accurately translate the given verbal information (distance between foci, minor axis length) into their corresponding standard ellipse parameters ($2ae$ and $2b$).
2. Utilize the derived values of $ae$ and $b$ in the most appropriate form of the fundamental relationship, $b^2 = a^2 - (ae)^2$, to find the semi-major axis $a$.
3. Finally, calculate the eccentricity $e$ using its definition $e = \frac{ae}{a}$.

A solid understanding of these core formulas and their interrelationships is absolutely essential for mastering conic sections problems in JEE Mathematics.