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 rectangular hyperbola and determine the conditions under which its branches lie only in the second and fourth quadrants.

### Step-by-Step Solution:

1. **Identify the Given Equation**:
The equation of the hyperbola is given as:
\[ xy = \lambda^2 - 9 \]

2. **Determine the Nature of the Hyperbola**:
We know that a rectangular hyperbola has the general form:
\[ xy = c \]
where \(c\) can be positive or negative. The branches of the hyperbola will lie in different quadrants depending on the sign of \(c\).

3. **Condition for Quadrants**:
The problem states that the branches of the hyperbola lie only in the second and fourth quadrants. This means that the value of \(c\) must be negative. Therefore, 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, we obtain:
\[ -3 < \lambda < 3 \]
or in terms of absolute value:
\[ |\lambda| < 3 \]

6. **Conclusion**:
The values of \(\lambda\) that satisfy the condition for the hyperbola's branches to lie only in the second and fourth quadrants are:
\[ \lambda \in (-3, 3) \]

### Final Answer:
The value of \(\lambda\) must satisfy:
\[ |\lambda| < 3 \]