If `xy = m^(2) -4` be a reactangular hyperbola whose branches lies only in the `2^(nd) and 4^(th)` quadrant then

Options

A. `|m| lt 2`
B. `|m|gt2`
C. `|m| lt 2`
D. not possible

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. **Understand the Equation**:
The equation given is \( xy = m^2 - 4 \). This represents a rectangular hyperbola.

2. **Identify the Quadrants**:
The branches of the hyperbola will lie in the second and fourth quadrants if the constant on the right side of the equation is negative. This is because:
- In the second quadrant, \( x < 0 \) and \( y > 0 \) (the product \( xy < 0 \)).
- In the fourth quadrant, \( x > 0 \) and \( y < 0 \) (the product \( xy < 0 \)).
Therefore, for the hyperbola to lie in these quadrants, we need \( m^2 - 4 < 0 \).

3. **Set Up the Inequality**:
From the condition \( m^2 - 4 < 0 \), we can rearrange this to:
\[ m^2 < 4 \]

4. **Solve the Inequality**:
Taking the square root of both sides, we find:
\[ |m| < 2 \]
This means that \( m \) must be in the range:
\[ -2 < m < 2 \]

5. **Conclusion**:
The condition for the branches of the hyperbola \( xy = m^2 - 4 \) to lie only in the second and fourth quadrants is that the absolute value of \( m \) must be less than 2.

### Final Answer:
Thus, the correct condition is:
\[ |m| < 2 \]