Understanding the Hyperbola: Essential Concepts and Formulas

To effectively solve problems involving hyperbolas, it's crucial to have a firm grasp of their fundamental definitions and standard formulas. A hyperbola is a conic section characterized by two distinct branches. For a hyperbola centered at the origin $(0,0)$ with its transverse axis along the x-axis (the most common standard form), its equation is: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Here, $a$ and $b$ are positive real numbers representing key dimensions:

• $a$: This is the length of the semi-transverse axis. The transverse axis is the line segment connecting the two vertices of the hyperbola, and its total length is $2a$.
• $b$: This is the length of the semi-conjugate axis. The conjugate axis is perpendicular to the transverse axis and has a total length of $2b$.
• $e$: This is the eccentricity of the hyperbola. For any hyperbola, the eccentricity $e$ is always greater than 1 ($e > 1$). It quantifies how "open" the branches of the hyperbola are; a larger $e$ means wider branches.

The specific formulas directly relevant to this problem are:

1. Length of the conjugate axis: This is given by $2b$.
2. Distance between the foci: The two foci (plural of focus) of the hyperbola are located at $(\pm ae, 0)$ for a transverse axis along the x-axis. Therefore, the distance between them is $2ae$.
3. Fundamental relationship between $a, b,$ and $e$: This identity connects the semi-transverse axis, semi-conjugate axis, and eccentricity. It is given by: $b^2 = a^2(e^2 - 1)$ This relationship can be algebraically rearranged into a very useful form by expanding: $b^2 = a^2e^2 - a^2$. Rearranging further, we get: $a^2e^2 = a^2 + b^2$ This form is often the most convenient as it directly relates the square of the distance from the center to a focus ($ae$) to $a^2$ and $b^2$.

Problem Analysis and Given Information

The question provides us with two pieces of information about a hyperbola and asks for its eccentricity:

• Length of its conjugate axis = 5
• Distance between its foci = 13

Our objective is to determine the value of $e$.

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Step-by-Step Solution

Step 1: Formulating Equations from the Given Information

We begin by translating the given verbal descriptions into mathematical equations using the standard formulas for a hyperbola.

• Using the length of the conjugate axis: The problem states that the length of the conjugate axis is 5. We know that the formula for the length of the conjugate axis is $2b$. Therefore, we can establish the equation: $2b = 5$ Why this step? By equating the given length to its standard formula, we can directly determine the value of $b$, the semi-conjugate axis. This parameter $b$ is one of the fundamental dimensions of the hyperbola and will be used in subsequent calculations. Dividing both sides by 2, we find $b$: $b = \frac{5}{2}$

• Using the distance between the foci: The problem states that the distance between the foci is 13. We know that the formula for the distance between the foci is $2ae$. Therefore, we set up the equation: $2ae = 13$ Why this step? Similar to the previous step, we equate the given distance to its standard formula. This gives us the product $ae$, which represents the distance from the center of the hyperbola to each focus. This product is crucial because it directly involves the eccentricity $e$ that we need to find. Dividing both sides by 2, we find the value of $ae$: $ae = \frac{13}{2}$

Step 2: Utilizing the Fundamental Relationship to Find the Semi-Transverse Axis ($a$)

Now we have values for $b$ and $ae$. Our goal is to find $e$. To do this, it's often easiest to first find $a$, the semi-transverse axis. The fundamental relationship $a^2e^2 = a^2 + b^2$ is the key to connecting these parameters.

Why this step? This identity is crucial because it links $a$, $b$, and $e$. We have values for $b$ and the product $ae$. By substituting these known values into the identity, we can form an equation with only $a^2$ as the unknown, allowing us to solve for $a$.

First, let's square the value of $b$ from Step 1: $b^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4}$ Next, let's square the value of $ae$ from Step 1: $(ae)^2 = \left(\frac{13}{2}\right)^2$ $a^2e^2 = \frac{169}{4}$ Why square these values? The fundamental relationship uses $b^2$ and $a^2e^2$, so squaring our known values of $b$ and $ae$ directly prepares them for substitution into the identity.

Now, substitute $b^2 = \frac{25}{4}$ and $a^2e^2 = \frac{169}{4}$ into the fundamental relationship $a^2e^2 = a^2 + b^2$: $\frac{169}{4} = a^2 + \frac{25}{4}$ Why this substitution? By substituting the numerical values for $b^2$ and $a^2e^2$, we obtain an algebraic equation that contains only $a^2$ as an unknown, making it solvable for $a^2$.

Now, we solve for $a^2$: $a^2 = \frac{169}{4} - \frac{25}{4}$ $a^2 = \frac{169 - 25}{4}$ $a^2 = \frac{144}{4}$ $a^2 = 36$ To find $a$, we take the square root of $a^2$: $a = \sqrt{36}$ $a = 6$ Why take the positive square root? Since $a$ represents a length (the semi-transverse axis), it must always be a positive value.

Step 3: Calculating the Eccentricity ($e$)

We now have all the necessary information to find the eccentricity $e$. From Step 1, we know $ae = \frac{13}{2}$, and from Step 2, we just found $a = 6$.

Why this step? With the value of $a$ now known, we can directly substitute it back into the equation involving $ae$ to isolate and solve for $e$. This is the final step to achieve our objective.

Substitute the value of $a=6$ into the equation $ae = \frac{13}{2}$: $(6)e = \frac{13}{2}$ Finally, solve for $e$: $e = \frac{13}{2 \times 6}$ $e = \frac{13}{12}$

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Verification and Self-Check

For a hyperbola, the eccentricity $e$ must always be greater than 1 ($e > 1$). Our calculated value $e = \frac{13}{12}$ is approximately $1.083...$. Since $1.083... > 1$, our result is consistent with the properties of a hyperbola, which provides a good check on our calculations.

The final answer is $\boxed{\text{13/12}}$.

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JEE Tips and Common Mistakes to Avoid

• Master the Formulas: It is absolutely essential to memorize the definitions and formulas for the lengths of axes, focal distances, and the fundamental relationship ($b^2 = a^2(e^2 - 1)$ or $a^2e^2 = a^2 + b^2$) for a hyperbola.
• Hyperbola vs. Ellipse: A very common and critical mistake is to confuse the fundamental relationship for a hyperbola with that of an ellipse.
  • For an ellipse: $b^2 = a^2(1 - e^2)$ (or $a^2e^2 = a^2 - b^2$), where $e < 1$.
  • For a hyperbola: $b^2 = a^2(e^2 - 1)$ (or $a^2e^2 = a^2 + b^2$), where $e > 1$. Notice the crucial sign difference and the condition on $e$.
• Positive Values for Lengths: Remember that $a$ and $b$ represent lengths (semi-axes) and must always be positive. If your calculations yield a negative value for $a$ or $b$, recheck your steps.
• Eccentricity Range: Always verify that your calculated eccentricity for a hyperbola is greater than 1. This serves as a quick self-check to catch errors.
• Algebraic Precision: Pay close attention to arithmetic operations, especially when dealing with fractions and squares. A small calculation error can lead to an incorrect final answer.

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Summary and Key Takeaway

This problem is a classic application of the standard definitions and relationships of a hyperbola to find its eccentricity. The methodical approach involves:

1. Identifying Given Information: Translate verbal problem statements into mathematical equations using standard formulas ($2b=5$, $2ae=13$).
2. Solving for Parameters: Use these equations to find individual parameters like $b$ and the product $ae$.
3. Applying the Fundamental Identity: Leverage the core identity $a^2e^2 = a^2 + b^2$ (or $b^2 = a^2(e^2 - 1)$) to solve for the semi-transverse axis, $a$.
4. Calculating Eccentricity: Finally, use the values of $a$ and $ae$ to determine $e$.

The ability to fluidly use these standard formulas and relationships is a core skill for solving problems involving conic sections in JEE Mathematics.