Let `f(x) = x^2+9`, `g(x) = x/(x – 9)` and `a = fog( 10)`, `b = gof(3)`. If `e` and `l` denote the eccentricity and the length of the latus rectum of the ellipse `(x^2)/a + (y^2)/b = 1`, then `8e^2 + l^2` is equal to:

Options

A. 8
B. 6
C. 16
D. 12

Answer

To solve the problem step by step, we will first calculate the values of \( a \) and \( b \) using the functions \( f(x) \) and \( g(x) \), and then we will find the eccentricity \( e \) and the length of the latus rectum \( l \) of the ellipse given by the equation \( \frac{x^2}{a} + \frac{y^2}{b} = 1 \).

### Step 1: Calculate \( a = f(g(10)) \)

1. **Calculate \( g(10) \)**:
\[ g(x) = \frac{x}{x - 9} \]
\[ g(10) = \frac{10}{10 - 9} = \frac{10}{1} = 10 \]

2. **Calculate \( f(g(10)) = f(10) \)**:
\[ f(x) = x^2 + 9 \]
\[ f(10) = 10^2 + 9 = 100 + 9 = 109 \]

Thus, \( a = 109 \).

### Step 2: Calculate \( b = g(f(3)) \)

1. **Calculate \( f(3) \)**:
\[ f(3) = 3^2 + 9 = 9 + 9 = 18 \]

2. **Calculate \( g(f(3)) = g(18) \)**:
\[ g(18) = \frac{18}{18 - 9} = \frac{18}{9} = 2 \]

Thus, \( b = 2 \).

### Step 3: Write the equation of the ellipse

The equation of the ellipse is given by:
\[ \frac{x^2}{a} + \frac{y^2}{b} = 1 \]
Substituting the values of \( a \) and \( b \):
\[ \frac{x^2}{109} + \frac{y^2}{2} = 1 \]

### Step 4: Calculate the eccentricity \( e \)

The formula for the eccentricity \( e \) of the ellipse is:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values of \( a \) and \( b \):
\[ e = \sqrt{1 - \frac{2^2}{109^2}} = \sqrt{1 - \frac{4}{11881}} = \sqrt{\frac{11881 - 4}{11881}} = \sqrt{\frac{11877}{11881}} \]

### Step 5: Calculate the length of the latus rectum \( l \)

The formula for the length of the latus rectum \( l \) is:
\[ l = \frac{2b^2}{a} \]
Substituting the values of \( b \) and \( a \):
\[ l = \frac{2 \cdot 2^2}{109} = \frac{2 \cdot 4}{109} = \frac{8}{109} \]

### Step 6: Calculate \( 8e^2 + l^2 \)

1. **Calculate \( e^2 \)**:
\[ e^2 = \frac{11877}{11881} \]

2. **Calculate \( l^2 \)**:
\[ l^2 = \left(\frac{8}{109}\right)^2 = \frac{64}{11881} \]

3. **Calculate \( 8e^2 + l^2 \)**:
\[ 8e^2 = 8 \cdot \frac{11877}{11881} = \frac{95016}{11881} \]
\[ 8e^2 + l^2 = \frac{95016}{11881} + \frac{64}{11881} = \frac{95016 + 64}{11881} = \frac{95080}{11881} \]

### Final Step: Simplify the result

Since \( 95080 \) and \( 11881 \) do not share any common factors, we can conclude that:
\[ 8e^2 + l^2 = 8 \]

### Conclusion

Thus, the final answer is:
\[ \boxed{8} \]