If `(x^(2))/(lambda+3)+(y^(2))/(2-lambda)=1` represents a hyperbola then

Options

A. `lambda in (2,3)`
B. `lambda in (-2,3)`
C. `lambda in (-3,2)`
D. `lambda in (2,infty)`

Answer

To determine the range of \(\lambda\) for which the equation

\[ \frac{x^2}{\lambda + 3} + \frac{y^2}{2 - \lambda} = 1 \]

represents a hyperbola, we need to analyze the conditions under which this equation holds true.

### Step 1: Identify the conditions for a hyperbola
For the given equation to represent a hyperbola, the denominators of the fractions must have opposite signs. This means that one must be positive and the other must be negative.

### Step 2: Set up inequalities
We have two fractions:
1. \(\lambda + 3\)
2. \(2 - \lambda\)

We need to establish conditions for these two expressions:

1. \(\lambda + 3 > 0\) (This implies \(\lambda > -3\))
2. \(2 - \lambda < 0\) (This implies \(\lambda > 2\))

### Step 3: Analyze the inequalities
From the first inequality, we have:
\[ \lambda > -3 \]
From the second inequality, we have:
\[ \lambda > 2 \]

### Step 4: Combine the conditions
Since both conditions must be satisfied simultaneously for the equation to represent a hyperbola, we take the more restrictive condition:
\[ \lambda > 2 \]

### Step 5: Consider the opposite case
Now, let's consider the opposite case where:
1. \(\lambda + 3 < 0\) (This implies \(\lambda < -3\))
2. \(2 - \lambda > 0\) (This implies \(\lambda < 2\))

Combining these conditions gives:
\[ \lambda < -3 \]

### Step 6: Final range of \(\lambda\)
Thus, the complete range of \(\lambda\) for which the equation represents a hyperbola is:
\[ \lambda < -3 \quad \text{or} \quad \lambda > 2 \]

### Conclusion
The final answer can be expressed in interval notation as:
\[ \lambda \in (-\infty, -3) \cup (2, \infty) \]