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 it can be classified as either an ellipse or a hyperbola.

### Step-by-Step Solution:

1. **Identify the Form of the Equation**:
The given equation is of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), which is the standard form of an ellipse if both denominators are positive.

2. **Set Conditions for Ellipse**:
For the equation to represent an ellipse:
- \(12 - \lambda > 0\) (i.e., \(12 > \lambda\) or \(\lambda < 12\))
- \(8 - \lambda > 0\) (i.e., \(8 > \lambda\) or \(\lambda < 8\))

3. **Combine Conditions for Ellipse**:
The more restrictive condition is \(\lambda < 8\). Therefore, for the equation to represent an ellipse, \(\lambda\) must be less than 8.

4. **Check for Hyperbola**:
The equation can also represent a hyperbola if one of the denominators is negative.
- If \(12 - \lambda > 0\) (i.e., \(\lambda < 12\)) and \(8 - \lambda < 0\) (i.e., \(\lambda > 8\)), then the equation can represent a hyperbola.
- This gives us the condition \(8 < \lambda < 12\).

5. **Conclusion**:
- If \(\lambda < 8\), the equation represents an ellipse.
- If \(8 < \lambda < 12\), the equation represents a hyperbola.
- If \(\lambda > 12\), then \(12 - \lambda < 0\) and \(8 - \lambda < 0\), which does not satisfy the conditions for either an ellipse or a hyperbola.

Thus, the equation \(\frac{x^2}{12 - \lambda} + \frac{y^2}{8 - \lambda} = 1\) represents:
- An ellipse for \(\lambda < 8\)
- A hyperbola for \(8 < \lambda < 12\)

### Final Answer:
The equation represents an ellipse when \(\lambda < 8\) and a hyperbola when \(8 < \lambda < 12\).