The equation `(x^2)/(12-lambda)+(y^2)/(8-lambda)=1` represents

Options

A. a hyperbola if `lambda lt 8`
B. an ellipse if `lambda gt 0`
C. a hyperbola if `8 lt lambda lt 12`
D. None of these

Answer

To determine what the equation \(\frac{x^2}{12 - \lambda} + \frac{y^2}{8 - \lambda} = 1\) represents, we need to analyze the conditions under which the denominators are positive or negative.

### Step-by-Step Solution:

1. **Identify the Form of the Equation**:
The given equation is of the form \(\frac{x^2}{a} + \frac{y^2}{b} = 1\), where \(a = 12 - \lambda\) and \(b = 8 - \lambda\).

2. **Determine Conditions for \(a\) and \(b\)**:
- For the equation to represent an ellipse, both \(a\) and \(b\) must be positive.
- For the equation to represent a hyperbola, one of \(a\) or \(b\) must be negative, while the other must be positive.

3. **Analyze the Conditions**:
- **Case 1**: If \(\lambda < 8\):
- Then \(12 - \lambda > 0\) (since \(12 - \lambda\) will be positive).
- \(8 - \lambda > 0\) (since we are subtracting a value less than 8 from 8).
- Thus, both \(a\) and \(b\) are positive, and the equation represents an ellipse.

- **Case 2**: If \(0 < \lambda < 8\):
- \(12 - \lambda > 0\) (still positive).
- \(8 - \lambda > 0\) (still positive).
- The equation still represents an ellipse.

- **Case 3**: If \(8 < \lambda < 12\):
- \(12 - \lambda > 0\) (positive).
- \(8 - \lambda < 0\) (negative).
- Here, the equation takes the form \(\frac{x^2}{\text{positive}} - \frac{y^2}{\text{positive}} = 1\), which represents a hyperbola.

- **Case 4**: If \(\lambda \geq 12\):
- \(12 - \lambda \leq 0\) (non-positive).
- The equation does not represent a standard conic section.

4. **Conclusion**:
- The equation represents an ellipse when \(\lambda < 8\).
- The equation represents a hyperbola when \(8 < \lambda < 12\).
- Therefore, the correct answer is that the equation represents a hyperbola when \(\lambda\) is between 8 and 12.

### Final Answer:
The equation \(\frac{x^2}{12 - \lambda} + \frac{y^2}{8 - \lambda} = 1\) represents a hyperbola when \(8 < \lambda < 12\).