This problem is a multi-concept question designed to test your understanding of composite functions, the standard form of an ellipse, and its key properties like eccentricity and the length of the latus rectum. We will systematically break down the problem into manageable steps, explaining the 'what' and 'why' behind each calculation.

---

1. Core Concepts and Formulas for Success

Before we dive into the calculations, let's revisit the fundamental definitions and formulas essential for solving this problem. A strong grasp of these will ensure accuracy and efficiency.

• Composite Functions: Given two functions $f(x)$ and $g(x)$:

  • The composition $f \circ g(x)$ is defined as $f(g(x))$. This means we first evaluate the inner function $g(x)$ and then use that result as the input for the outer function $f(x)$.
  • Similarly, $g \circ f(x)$ is defined as $g(f(x))$. Here, $f(x)$ is evaluated first, and its output becomes the input for $g(x)$.
  • Key Tip: Always evaluate composite functions from the "inside out." The order of composition is crucial; $f \circ g(x)$ is generally not equal to $g \circ f(x)$.

• Standard Ellipse Equation: An ellipse centered at the origin has the standard form: $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$

  • Here, $A^2$ and $B^2$ are positive constants representing the squares of the semi-axes lengths along the x and y directions, respectively.
  • Crucial Identification: To determine the ellipse's properties, we must identify the semi-major axis ($A_{maj}$) and the semi-minor axis ($A_{min}$).
    • $A_{maj}^2$ is always the larger of the two denominators ($A^2$ or $B^2$).
    • $A_{min}^2$ is always the smaller of the two denominators ($A^2$ or $B^2$).
    • If $A^2 > B^2$, then $A_{maj}^2 = A^2$ (major axis along x-axis).
    • If $B^2 > A^2$, then $A_{maj}^2 = B^2$ (major axis along y-axis).
  • Common Mistake Alert: Be extremely careful not to confuse the variables 'a' and 'b' given in the problem statement (which are just parameters for the ellipse) with the standard notation 'a' and 'b' often used for semi-major and semi-minor axes in textbooks. To avoid this, we will consistently use $A_{maj}$ and $A_{min}$ for the semi-major and semi-minor axis lengths, respectively.

• Eccentricity ($e$): Eccentricity quantifies how "stretched" or "flat" an ellipse is. For an ellipse, its value always lies between 0 and 1 (exclusive). The formula is: $e = \sqrt{1 - \frac{A_{min}^2}{A_{maj}^2}}$ It is often more convenient to work with $e^2$: $e^2 = 1 - \frac{A_{min}^2}{A_{maj}^2}$

  • Why this formula? This formula directly relates the semi-axes to the ellipse's shape. It stems from the geometric definition of an ellipse, where the distance from the center to a focus $c$ is related by $c^2 = A_{maj}^2 - A_{min}^2$, and eccentricity is defined as $e = c/A_{maj}$.

• Length of the Latus Rectum ($l$): The latus rectum is a chord that passes through a focus of the ellipse and is perpendicular to the major axis. Its length is given by: $l = \frac{2 A_{min}^2}{A_{maj}}$

  • Why this formula? This formula is derived from the coordinate geometry of an ellipse, specifically by finding the y-coordinate of the points on the ellipse when $x = \pm c$ (where $c$ is the focal distance).

---

2. Step 1: Evaluating Composite Functions to Determine 'a' and 'b'

Our first objective is to calculate the specific numerical values of 'a' and 'b' using the given functions $f(x)=x^2+9$ and $g(x)=\frac{x}{x-9}$. These values will then define our ellipse.

2.1 Calculating $\mathrm{a} = f \circ g(10)$

To find $f \circ g(10)$, we apply the definition $f(g(10))$. This means we must first evaluate the inner function, $g(x)$, at $x=10$.

• Why? The output of $g(10)$ will serve as the input for the function $f(x)$.

Let's compute $g(10)$: $g(10) = \frac{10}{10-9} = \frac{10}{1} = 10$ Now, we substitute this result into the outer function $f(x)$. So, $\mathrm{a} = f(g(10))$ becomes $f(10)$.

• Why? We've found that $g(10)$ evaluates to $10$, so we are now evaluating $f$ at this specific value.

Let's compute $f(10)$: $f(10) = (10)^2 + 9 = 100 + 9 = 109$ Thus, the value of $\mathrm{a} = 109$.

2.2 Calculating $\mathrm{b} = g \circ f(3)$

Similarly, to find $g \circ f(3)$, we apply the definition $g(f(3))$. We must first evaluate the inner function, $f(x)$, at $x=3$.

• Why? The output of $f(3)$ will become the input for the function $g(x)$.

Let's compute $f(3)$: $f(3) = (3)^2 + 9 = 9 + 9 = 18$ Next, we substitute this result into the outer function $g(x)$. So, $\mathrm{b} = g(f(3))$ becomes $g(18)$.

• Why? We've determined that $f(3)$ evaluates to $18$, so we are now evaluating $g$ at this value.

Let's compute $g(18)$: $g(18) = \frac{18}{18-9} = \frac{18}{9} = 2$ Thus, the value of $\mathrm{b} = 2$.

Summary for Step 1: We have successfully determined the parameters for our ellipse: $\mathrm{a} = 109$ and $\mathrm{b} = 2$.

---

3. Step 2: Defining the Ellipse and Identifying its Axes

Now we use the calculated values of $\mathrm{a}$ and $\mathrm{b}$ to form the specific equation of the ellipse and then identify its semi-major and semi-minor axes.

3.1 Forming the Specific Ellipse Equation

The problem states the ellipse equation is $\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{~b}}=1$. Substitute the values $\mathrm{a}=109$ and $\mathrm{b}=2$ into this equation: $\frac{x^2}{109} + \frac{y^2}{2} = 1$

3.2 Determining Semi-major and Semi-minor Axes

We compare our specific ellipse equation with the standard form $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$. From the comparison, we identify:

• $A^2 = 109$

• $B^2 = 2$

• Why is this comparison crucial? To correctly calculate the eccentricity and latus rectum, we need to distinguish between the semi-major axis ($A_{maj}$) and the semi-minor axis ($A_{min}$). These are determined by the larger and smaller of the denominators, respectively.

Since $109 > 2$, the semi-major axis squared is $A_{maj}^2 = 109$, and the semi-minor axis squared is $A_{min}^2 = 2$.

• This also tells us that the major axis of this ellipse lies along the x-axis.

From these squared values, we can find the actual semi-axis lengths:

• $A_{maj} = \sqrt{109}$
• $A_{min} = \sqrt{2}$

Important Reminder: We are using $A_{maj}$ and $A_{min}$ to avoid confusion with the problem's 'a' and 'b' variables.

---

4. Step 3: Calculating the Eccentricity ($e$)

Now that we have identified $A_{maj}^2$ and $A_{min}^2$, we can calculate the eccentricity $\mathrm{e}$.

4.1 Formula for Eccentricity Squared

We will use the formula for $e^2$: $e^2 = 1 - \frac{A_{min}^2}{A_{maj}^2}$

• Why this form? The final expression requires $e^2$, so calculating $e^2$ directly avoids an unnecessary square root operation and potential rounding errors.

4.2 Substitution and Calculation of $\mathrm{e}^2$

Substitute $A_{maj}^2 = 109$ and $A_{min}^2 = 2$ into the formula: $e^2 = 1 - \frac{2}{109}$ To simplify, find a common denominator: $e^2 = \frac{109}{109} - \frac{2}{109}$ $e^2 = \frac{109 - 2}{109}$ $e^2 = \frac{107}{109}$

---

5. Step 4: Calculating the Length of the Latus Rectum ($l$)

Next, we calculate the length of the latus rectum $l$.

5.1 Formula for Latus Rectum

The formula for the length of the latus rectum $l$ is: $l = \frac{2 A_{min}^2}{A_{maj}}$

• Why this formula? This is a standard formula derived from the geometric properties of an ellipse, crucial for describing its shape.

5.2 Substitution and Calculation of $l$ and $l^2$

Substitute $A_{min}^2 = 2$ and $A_{maj} = \sqrt{109}$ into the formula: $l = \frac{2 \times 2}{\sqrt{109}}$ $l = \frac{4}{\sqrt{109}}$ For the final expression, we will need $l^2$. Let's calculate that now: $l^2 = \left(\frac{4}{\sqrt{109}}\right)^2$ $l^2 = \frac{4^2}{(\sqrt{109})^2}$ $l^2 = \frac{16}{109}$

---

6. Step 5: Final Calculation of $8\mathrm{e}^2 + l^2$

We have successfully calculated $\mathrm{e}^2 = \frac{107}{109}$ and $l^2 = \frac{16}{109}$. Now, we can substitute these values into the expression $8\mathrm{e}^2 + l^2$.

6.1 Substituting Calculated Values

$8\mathrm{e}^2 + l^2 = 8 \left(\frac{107}{109}\right) + \frac{16}{109}$

6.2 Simplification

First, multiply 8 by $\frac{107}{109}$: $8 \times 107 = 856$ So the expression becomes: $8\mathrm{e}^2 + l^2 = \frac{856}{109} + \frac{16}{109}$ Since both terms have the same denominator, we can simply add the numerators: $8\mathrm{e}^2 + l^2 = \frac{856 + 16}{109}$ $8\mathrm{e}^2 + l^2 = \frac{872}{109}$ Now, perform the division:

$$$$