If `xy=lambda^(2)-9` be a rectangular hyperbola whose branches lie only in the second and fourth quadrant, then

Options

A. `|lambda|ge3`
B. `|lambda|lt3`
C. `lambdainR-{-3, 3}`
D. None of these

Answer

To solve the problem, we need to analyze the given equation of the hyperbola and the conditions under which its branches lie only in the second and fourth quadrants.

### Step-by-Step Solution:

1. **Identify the Given Equation**:
We start with the equation of the hyperbola given by:
\[ xy = \lambda^2 - 9 \]

2. **Understand the Condition for Quadrants**:
The branches of the hyperbola lie in the second and fourth quadrants. For this to happen, the equation must yield negative values for \(xy\) in these quadrants.

3. **Set Up the Inequality**:
Since we want the branches to lie in the second and fourth quadrants, we need:
\[ \lambda^2 - 9 < 0 \]

4. **Solve the Inequality**:
Rearranging the inequality gives:
\[ \lambda^2 < 9 \]

5. **Find the Range for \(\lambda\)**:
Taking the square root of both sides leads us to:
\[ -3 < \lambda < 3 \]
This can be expressed in terms of absolute value:
\[ |\lambda| < 3 \]

6. **Conclusion**:
Thus, the condition for \(\lambda\) such that the branches of the hyperbola lie only in the second and fourth quadrants is:
\[ |\lambda| < 3 \]