If `x^(2)/(12-k) -(y^(2))/(k-8)=1` represents a hyperbola then

Options

A. a) `k lt 8`
B. b) `k gt 12`
C. c) `8 lt k lt 12`
D. d) `k= pm 12, k = +8`

Answer

To determine the values of \( k \) for which the equation

\[ \frac{x^2}{12-k} - \frac{y^2}{k-8} = 1 \]

represents a hyperbola, we need to ensure that both denominators are positive. This leads us to the following steps:

### Step 1: Identify the conditions for the denominators

The equation of the hyperbola can be rewritten in the standard form:

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

where \( a^2 = 12 - k \) and \( b^2 = k - 8 \). For the hyperbola to be valid, both \( a^2 \) and \( b^2 \) must be greater than zero.

### Step 2: Set up the inequalities

From the conditions \( a^2 > 0 \) and \( b^2 > 0 \), we derive the following inequalities:

1. \( 12 - k > 0 \)
2. \( k - 8 > 0 \)

### Step 3: Solve the inequalities

1. For the first inequality \( 12 - k > 0 \):
\[ -k > -12 \implies k < 12 \]

2. For the second inequality \( k - 8 > 0 \):
\[ k > 8 \]

### Step 4: Combine the results

Combining both results, we find:

\[ 8 < k < 12 \]

### Conclusion

The values of \( k \) for which the given equation represents a hyperbola are:

\[ k \in (8, 12) \]