This solution will guide you through a typical JEE problem involving the properties of a hyperbola. We will start by revisiting the fundamental definitions and formulas, then systematically apply them to the given information, solve the resulting system of equations, and finally arrive at the required value.

---

1. Fundamental Concepts of a Hyperbola

A hyperbola is a conic section characterized by its unique geometric properties. For a standard hyperbola centered at the origin, with its transverse axis lying along the x-axis, its equation is given by: $H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Here, $a > 0$ and $b > 0$ are constants that define the shape and size of the hyperbola. Let's recall the key properties essential for this problem:

• Length of the Transverse Axis: This is the distance between the two vertices of the hyperbola. Its length is $2a$.
• Length of the Conjugate Axis: This axis is perpendicular to the transverse axis and passes through the center. Its length is $2b$.
• Eccentricity ($e$): This parameter measures the "openness" or "flatness" of the hyperbola. For the standard form given, the eccentricity is related to $a$ and $b$ by the formula: $e^2 = 1 + \frac{b^2}{a^2}$ For any hyperbola, its eccentricity $e$ is always greater than 1 ($e > 1$).

Understanding these definitions and formulas is the first crucial step in translating the problem statement into solvable mathematical equations.

---

2. Translating Problem Information into Equations

The problem provides us with two distinct pieces of information about the hyperbola $H$. Our goal in this section is to convert these verbal statements into algebraic equations involving $a$ and $b$.

• Information 1: Sum of the lengths of the transverse and conjugate axes. The problem states that this sum is $4(2\sqrt{2} + \sqrt{14})$.

  • Why this is important: We can directly apply the definitions of the axis lengths ($2a$ and $2b$) to form our first equation. $2a + 2b = 4(2\sqrt{2} + \sqrt{14})$ To simplify this equation and make it easier to work with, we can divide both sides by 2: $a + b = 2(2\sqrt{2} + \sqrt{14})$ $a + b = 4\sqrt{2} + 2\sqrt{14} \quad \text{...(1)}$ This equation establishes a linear relationship between $a$ and $b$.

• Information 2: Eccentricity of the hyperbola ($e$). We are given that the eccentricity $e = \frac{\sqrt{11}}{2}$.

  • Why this is important: We can use the fundamental eccentricity formula for a hyperbola to establish a second, independent relationship between $a$ and $b$. Substituting the given value of $e$ into the formula $e^2 = 1 + \frac{b^2}{a^2}$: $\left(\frac{\sqrt{11}}{2}\right)^2 = 1 + \frac{b^2}{a^2}$ Squaring the left side gives: $\frac{11}{4} = 1 + \frac{b^2}{a^2} \quad \text{...(2)}$ This equation provides a quadratic relationship between $a$ and $b$.

Now we have a system of two equations (Equation 1 and Equation 2) with two unknowns ($a$ and $b$). Our next step is to solve this system to find the individual values of $a$ and $b$, and then compute $a^2 + b^2$.

---

3. Step-by-Step Solution to Find $a^2 + b^2$

Let's systematically solve the system of equations derived in the previous section.

Step 3.1: Establish a direct relationship between $a$ and $b$ from the eccentricity equation. We start with Equation (2): $\frac{11}{4} = 1 + \frac{b^2}{a^2}$

• Why this step is taken: Our goal is to express $b$ in terms of $a$ (or vice-versa) to substitute into Equation (1). Isolating $\frac{b^2}{a^2}$ is the first step towards this. Subtract 1 from both sides of the equation: $\frac{b^2}{a^2} = \frac{11}{4} - 1$ $\frac{b^2}{a^2} = \frac{11 - 4}{4}$ $\frac{b^2}{a^2} = \frac{7}{4}$ Now, to find the ratio $\frac{b}{a}$, we take the square root of both sides. Since $a$ and $b$ represent lengths, they must be positive values, so we only consider the positive square root. $\frac{b}{a} = \sqrt{\frac{7}{4}}$ $\frac{b}{a} = \frac{\sqrt{7}}{2}$ This gives us $b$ in terms of $a$: $b = \frac{\sqrt{7}}{2}a \quad \text{...(3)}$
• Explanation: This simplified relationship (Equation 3) is now ready to be substituted into Equation (1), which will allow us to solve for $a$ (or $b$) directly.

Step 3.2: Substitute the relationship from Equation (3) into Equation (1) to solve for $a$. Recall Equation (1): $a + b = 4\sqrt{2} + 2\sqrt{14}$.

• Why this step is taken: By substituting $b$ in terms of $a$, we transform Equation (1) into an equation with only one unknown, $a$, making it solvable. Substitute $b = \frac{\sqrt{7}}{2}a$ from Equation (3) into Equation (1): $a + \frac{\sqrt{7}}{2}a = 4\sqrt{2} + 2\sqrt{14}$ Factor out $a$ from the left side of the equation: $a\left(1 + \frac{\sqrt{7}}{2}\right) = 4\sqrt{2} + 2\sqrt{14}$ Combine the terms inside the parenthesis on the left side by finding a common denominator: $a\left(\frac{2 + \sqrt{7}}{2}\right) = 4\sqrt{2} + 2\sqrt{14}$

Step 3.3: Simplify the right-hand side and solve for $a$. Let's simplify the right-hand side of the equation: $4\sqrt{2} + 2\sqrt{14}$.

• Why this step is taken: Simplifying radical expressions often reveals common factors that can lead to significant algebraic simplification, as we will see. We can rewrite $\sqrt{14}$ as $\sqrt{2 \times 7} = \sqrt{2}\sqrt{7}$. So, the right-hand side becomes: $4\sqrt{2} + 2\sqrt{2}\sqrt{7}$ Now, we can factor out the common term $2\sqrt{2}$: $2\sqrt{2}(2 + \sqrt{7})$ Substitute this simplified expression back into our equation from Step 3.2: $a\left(\frac{2 + \sqrt{7}}{2}\right) = 2\sqrt{2}(2 + \sqrt{7})$ Notice that the term $(2 + \sqrt{7})$ appears on both sides of the equation. Since $2 + \sqrt{7}$ is a non-zero value, we can divide both sides by $(2 + \sqrt{7})$: $\frac{a}{2} = 2\sqrt{2}$ Finally, multiply both sides by 2 to solve for $a$: $a = 4\sqrt{2}$
• Explanation: This cancellation is a powerful algebraic technique that significantly simplifies the equation and allows for direct calculation of $a$.

Step 3.4: Calculate $a^2$ and $b^2$. Now that we have the value of $a$, we can easily find $a^2$: $a^2 = (4\sqrt{2})^2 = 4^2 \times (\sqrt{2})^2 = 16 \times 2 = 32$ Next, we can use the relationship $\frac{b^2}{a^2} = \frac{7}{4}$ from Step 3.1 to find $b^2$: $b^2 = \frac{7}{4}a^2$

• Why this step is taken: The problem asks for $a^2 + b^2$, so calculating $a^2$ and $b^2$ individually is a necessary intermediate step. Using the ratio $\frac{b^2}{a^2}$ is more efficient than calculating $b$ first and then squaring it. Substitute the value of $a^2 = 32$: $b^2 = \frac{7}{4} \times 32$ $b^2 = 7 \times 8$ $b^2 = 56$

Step 3.5: Calculate $a^2 + b^2$. Finally, sum the calculated values of $a^2$ and $b^2$: $a^2 + b^2 = 32 + 56$ $a^2 + b^2 = 88$

• Why this step is taken: This is the final step to answer the specific question posed by the problem.

---

4. Important Tips and Common Pitfalls

• Eccentricity Formula: Always double-check the eccentricity formula based on the orientation of the hyperbola. For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (transverse axis along x-axis), it's $e^2 = 1 + \frac{b^2}{a^2}$. If the hyperbola were $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ (transverse axis along y-axis), the formula would be $e^2 = 1 + \frac{a^2}{b^2}$. A common mistake is to interchange $a$ and $b$ in the formula.
• Algebraic Simplification: Be very careful and meticulous with algebraic manipulations, especially when dealing with square roots and fractions. Factoring out common terms (like $2\sqrt{2}$ and $2 + \sqrt{7}$ in this problem) is a powerful technique that can dramatically simplify calculations and prevent errors.
• Positive Lengths: Remember that $a$ and $b$ represent lengths of semi-axes, so they must always be positive values. When taking square roots in the context of geometric dimensions, only consider the positive root. This is why we concluded $\frac{b}{a} = \frac{\sqrt{7}}{2}$ and not $\pm \frac{\sqrt{7}}{2}$.
• Systematic Approach: Break down complex problems into smaller, manageable steps. First, understand the concepts; second, translate the problem into equations; third, solve the system; and fourth, calculate the final answer. This systematic approach reduces the chances of errors and makes the solution process clearer.

---

5. Summary and Key Takeaway

This problem serves as an excellent exercise in applying the fundamental definitions and formulas of a hyperbola. The solution demonstrates a clear strategy for solving such problems:

1. Translate: Convert all given verbal information into precise mathematical equations using the definitions of hyperbola parameters.
2. Formulate: Set up a system of equations based on these translations.
3. Solve Systematically: Use substitution or other algebraic methods to solve for the unknown parameters ($a$ and $b$ in this case).
4. Simplify and Calculate: Perform careful algebraic manipulations, especially with radical expressions, to simplify intermediate results and arrive at the final answer.

By mastering these steps and being mindful of common pitfalls, you can confidently tackle similar problems in conic sections.

The final answer is $\boxed{88}$.