The equation `(x^(2))/(7-K) +(y^(2))/(5-K)=1` represents a hyperbola if

Options

A. A: `5 lt K lt 7`
B. B: `K lt 5 or K gt 7`
C. C: `K gt 5`
D. D: `K ne 5, K ne 7`

Answer

To determine the conditions under which the equation

\[ \frac{x^2}{7 - k} + \frac{y^2}{5 - k} = 1 \]

represents a hyperbola, we need to analyze the terms involved.

### Step-by-Step Solution:

1. **Identify the Standard Form of Hyperbola:**
The standard form of a hyperbola is given by:

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

where \(a^2 > 0\) and \(b^2 > 0\).

2. **Rearranging the Given Equation:**
The given equation can be rearranged to match the standard form of a hyperbola. We rewrite it as:

\[ \frac{x^2}{7 - k} - \frac{y^2}{k - 5} = 1 \]

Here, we can see that \(a^2 = 7 - k\) and \(b^2 = k - 5\).

3. **Setting Conditions for \(a^2\) and \(b^2\):**
For the equation to represent a hyperbola, both \(a^2\) and \(b^2\) must be positive:

- \(7 - k > 0\)
- \(k - 5 > 0\)

4. **Solving the Inequalities:**
- From \(7 - k > 0\):
\[ k < 7 \]
- From \(k - 5 > 0\):
\[ k > 5 \]

5. **Combining the Inequalities:**
We combine the two inequalities:

\[ 5 < k < 7 \]

This means \(k\) must be greater than 5 and less than 7.

### Final Result:
Thus, the equation represents a hyperbola if:

\[ k \in (5, 7) \]