The equation `(x^(2))/(9-lambda)+(y^(2))/(4-lambda) =1` represents a hyperbola when `a lt lambda lt b` then `(b-a)=`

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the problem, we need to analyze the given equation of the hyperbola and determine the conditions under which it represents a hyperbola.

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \frac{x^2}{9 - \lambda} + \frac{y^2}{4 - \lambda} = 1 \]

2. **Rearrange the equation**:
We can rewrite the equation in a more recognizable form for hyperbolas:
\[ \frac{x^2}{9 - \lambda} - \frac{y^2}{\lambda - 4} = 1 \]
This shows that we have a hyperbola because of the minus sign between the two terms.

3. **Identify conditions for a hyperbola**:
For the equation to represent a hyperbola, both denominators must be positive:
- \(9 - \lambda > 0\)
- \(\lambda - 4 > 0\)

4. **Solve the inequalities**:
- From \(9 - \lambda > 0\):
\[ \lambda < 9 \]
- From \(\lambda - 4 > 0\):
\[ \lambda > 4 \]

5. **Combine the inequalities**:
Thus, we have:
\[ 4 < \lambda < 9 \]
Here, \(a = 4\) and \(b = 9\).

6. **Calculate \(b - a\)**:
Now we find \(b - a\):
\[ b - a = 9 - 4 = 5 \]

### Final Answer:
\[ b - a = 5 \]