Introduction to Key Concepts and Formulas for a Hyperbola

To effectively solve this problem, a solid understanding of the fundamental properties and formulas associated with a standard hyperbola is essential. A hyperbola is a type of conic section defined by the locus of points where the absolute difference of the distances to two fixed points (foci) is constant.

For a hyperbola centered at the origin with its transverse axis along the x-axis, its standard equation is: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Here:

• $a$ represents the length of the semi-transverse axis. The transverse axis is the axis that passes through the foci and vertices.
• $b$ represents the length of the semi-conjugate axis. The conjugate axis is perpendicular to the transverse axis, passing through the center.
• $e$ represents the eccentricity of the hyperbola. For a hyperbola, $e > 1$, which describes how "open" the hyperbola is.

Let's list the specific formulas that are directly relevant to the problem statement:

1. Length of the Latus Rectum (LLR): The latus rectum is a chord of the hyperbola that passes through a focus and is perpendicular to the transverse axis. Its length is given by: $\text{LLR} = \frac{2b^2}{a}$ Significance: This formula relates the lengths of the semi-axes and provides a measure of the hyperbola's width at its foci.

2. Length of the Conjugate Axis: This is the total length of the segment along the conjugate axis, passing through the center. Its length is: $\text{Length of Conjugate Axis} = 2b$ Significance: This directly uses the semi-conjugate axis length $b$.

3. Distance Between Foci: The foci are two fixed points on the transverse axis that define the hyperbola. If the coordinates of the foci are $(\pm ae, 0)$, then the distance between them is: $\text{Distance Between Foci} = 2ae$ Significance: This formula relates the semi-transverse axis length $a$ and the eccentricity $e$ to the separation of the foci.

4. Fundamental Identity for a Hyperbola: This is a crucial relationship connecting $a$, $b$, and $e$: $b^2 = a^2(e^2 - 1)$ Significance: This identity is derived from the geometric properties of the hyperbola and is indispensable for solving problems that involve all three parameters. It's vital to remember that for a hyperbola, $e > 1$, which ensures $e^2 - 1$ is positive, making $b^2$ positive.

Our strategy will be to translate the given verbal conditions into mathematical equations using these formulas and then solve the resulting system of equations to find the eccentricity $e$.

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Step-by-Step Solution

Step 1: Translate the first condition into a mathematical equation.

The problem states: "the length of the latus rectum is equal to $8$".

• We recall the formula for the length of the latus rectum: $\text{LLR} = \frac{2b^2}{a}$.
• Equating this formula to the given value, we get our first equation: $\frac{2b^2}{a} = 8$
• To simplify this equation, we can multiply both sides by $a$ and divide by $2$: $b^2 = 4a \quad \text{--- (1)}$ Explanation: This step directly applies the definition of the length of the latus rectum. By converting the first piece of information into a mathematical relationship between the semi-axes $a$ and $b$, we establish a foundational equation that will be used in conjunction with other formulas.

Step 2: Translate the second condition into a mathematical equation.

The problem states: "the length of its conjugate axis is equal to half of the distance between its foci".

• We know the length of the conjugate axis is $2b$.
• We know the distance between its foci is $2ae$.
• Therefore, we can write the second condition mathematically as: $2b = \frac{1}{2}(2ae)$
• Now, we simplify this equation: $2b = ae \quad \text{--- (2)}$ Explanation: This step translates the second verbal description into a concise mathematical equation involving $b$, $a$, and $e$. This provides a second crucial relationship, connecting the dimensions of the hyperbola (via $b$) to its eccentricity and semi-transverse axis length (via $ae$).

Step 3: Combine the conditions with the fundamental identity to eliminate variables.

Our ultimate goal is to find the eccentricity $e$. To achieve this, we need to eliminate the variables $a$ and $b$ from our system of equations. The fundamental identity for a hyperbola, $b^2 = a^2(e^2 - 1)$, is the key to linking these variables.

• From equation (2), we have $2b = ae$. To make this expression compatible with the fundamental identity (which involves $b^2$), we square both sides of equation (2): $(2b)^2 = (ae)^2$ $4b^2 = a^2e^2$ Explanation: Squaring equation (2) is a strategic move. It transforms the expression from $b$ to $b^2$, which directly appears in the fundamental identity $b^2 = a^2(e^2 - 1)$. This prepares the equation for a crucial substitution that will eliminate $b$.

• Now, we substitute the fundamental identity, $b^2 = a^2(e^2 - 1)$, into the equation $4b^2 = a^2e^2$: $4 \left( a^2(e^2 - 1) \right) = a^2e^2$ Explanation: This is a critical substitution. By replacing $b^2$ with its equivalent expression in terms of $a$ and $e$, we successfully eliminate $b$ from our system of equations. We are now left with a single equation containing only $a$ and $e$, which is a significant step towards solving for $e$. Note that equation (1) ($b^2 = 4a$) was not directly used in this specific substitution path, though it could be used in an alternative approach to solve for $a$ or $b$ if needed.

Step 4: Solve for the eccentricity $e$.

We now have a single equation solely in terms of $a$ and $e$: $4a^2(e^2 - 1) = a^2e^2$

• Since $a$ represents the length of the semi-transverse axis, it must be a positive value, meaning $a \neq 0$. Therefore, we can safely divide both sides of the equation by $a^2$ without losing any solutions: $4(e^2 - 1) = e^2$
• Distribute the $4$ on the left side of the equation: $4e^2 - 4 = e^2$
• Rearrange the terms to isolate $e^2$. Subtract $e^2$ from both sides and add $4$ to both sides: $4e^2 - e^2 = 4$ $3e^2 = 4$
• Solve for $e^2$: $e^2 = \frac{4}{3}$
• Finally, take the square root of both sides to find $e$. Since eccentricity is defined as a positive value, we take only the positive root: $e = \sqrt{\frac{4}{3}}$ $e = \frac{\sqrt{4}}{\sqrt{3}}$ $e = \frac{2}{\sqrt{3}}$ Explanation: This step involves straightforward algebraic manipulation to isolate and solve for $e$. We obtain the numerical value for the eccentricity. It's crucial to remember that for a hyperbola, its eccentricity $e$ must be greater than $1$. Our result, $e = \frac{2}{\sqrt{3}} \approx \frac{2}{1.732} \approx 1.1547$, is indeed greater than $1$, which confirms the validity of our answer for a hyperbola.

Comparing our result with the given options: (A) ${2 \over {\sqrt 3 }}$ (B) ${\sqrt 3 }$ (C) ${{4 \over 3}}$ (D) ${4 \over {\sqrt 3 }}$

Our calculated eccentricity matches option (A).

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Important Tips for Success and Common Pitfalls

• Master the Formulas: The most common pitfall in conic sections problems is not knowing or misremembering the standard formulas. Always start by clearly listing the relevant formulas.
• Hyperbola vs. Ellipse Identity: A frequent mistake is confusing the fundamental identity for a hyperbola ($b^2 = a^2(e^2 - 1)$) with that for an ellipse ($b^2 = a^2(1 - e^2)$). Remember that for a hyperbola, $e > 1$, so $e^2 - 1$ must be positive. For an ellipse, $e < 1$, so $1 - e^2$ must be positive. This distinction is critical.
• Algebraic Precision: Be meticulous with your algebraic manipulations. Small errors in squaring terms, distributing constants, or rearranging equations can lead to entirely incorrect results. Double-check each step.
• Check Your Answer's Validity: After finding the eccentricity $e$, always perform a quick conceptual check. If the problem specifies a hyperbola, $e$ must be greater than $1$. If your calculated value for $e$ is $1$ or less, it's a strong indicator that an error has occurred somewhere in your calculations or formula application. This self-check can save you from choosing an incorrect option.
• Understand "Why": Don't just memorize formulas; understand the geometric meaning behind them and why certain algebraic steps (like squaring both sides or dividing by $a^2$) are valid and useful.

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Summary and Key Takeaway

This problem is an excellent test of one's fundamental understanding of hyperbolas. By systematically translating the given descriptive conditions ("length of latus rectum is 8" and "length of conjugate axis is half of the distance between foci") into precise mathematical equations using standard formulas, and then strategically combining these with the crucial fundamental identity $b^2 = a^2(e^2 - 1)$, we were able to isolate and solve for the eccentricity $e$. The final result, $e = \frac{2}{\sqrt{3}}$, not only provides the answer but also satisfies the condition $e > 1$ for a hyperbola. This problem highlights the importance of a strong conceptual foundation in conic sections and the ability to apply formulas accurately and systematically through algebraic manipulation.