If the equation `(x^(2))/(9-k)+(y^(2))/(5-k)=1` represents an ellipse then

Options

A. `kin(5,9)`
B. `kin[5,9]`
C. `klt5`
D. `kgt5`

Answer

To determine the conditions under which the equation

\[ \frac{x^2}{9-k} + \frac{y^2}{5-k} = 1 \]

represents an ellipse, we need to analyze the denominators of the fractions involved.

### Step-by-Step Solution:

1. **Identify the General Form of an Ellipse:**
The general form of an ellipse is given by

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \( a^2 > 0 \) and \( b^2 > 0 \). This means both denominators must be positive.

2. **Set Up Inequalities:**
For our equation to represent an ellipse, we need:

- \( 9 - k > 0 \)
- \( 5 - k > 0 \)

3. **Solve the First Inequality:**
From the first inequality \( 9 - k > 0 \):

\[ 9 > k \quad \text{or} \quad k < 9 \]

4. **Solve the Second Inequality:**
From the second inequality \( 5 - k > 0 \):

\[ 5 > k \quad \text{or} \quad k < 5 \]

5. **Combine the Results:**
We need both conditions to be satisfied simultaneously. Therefore, we take the more restrictive condition:

\[ k < 5 \]

### Conclusion:
For the equation to represent an ellipse, the value of \( k \) must satisfy:

\[ k < 5 \]