If one of the lines of `my^(2)+(1-m^(2))xy-mx^(2)=0` is a bisector of the angle between the lines `xy=0`, then m is

Options

A. 1
B. 2
C. `-1//2`
D. -1

Explanation

To solve the problem, we need to find the value of \( m \) such that one of the lines represented by the equation \( my^2 + (1 - m^2)xy - mx^2 = 0 \) is a bisector of the angle between the lines represented by \( xy = 0 \).

### Step-by-Step Solution:

1. **Understanding the given equation**:
The equation \( my^2 + (1 - m^2)xy - mx^2 = 0 \) represents a pair of straight lines. We can rewrite this equation to identify the lines.

2. **Rearranging the equation**:
We can rearrange the equation as follows:
\[ my^2 + (1 - m^2)xy - mx^2 = 0 \]
This can be factored as:
\[ y(my + (1 - m^2)x) - mx^2 = 0 \]

3. **Factoring the equation**:
We can factor out \( y \) from the first two terms:
\[ y(my + (1 - m^2)x) = mx^2 \]
This gives us:
\[ y = mx \quad \text{and} \quad y = -\frac{1}{m}x \]
So the two lines are:
\[ L_1: y = mx \quad \text{and} \quad L_2: y = -\frac{1}{m}x \]

4. **Identifying the angle bisectors**:
The lines \( xy = 0 \) represent the x-axis \( (y = 0) \) and the y-axis \( (x = 0) \). The angle bisectors of these lines are given by:
\[ y = x \quad \text{and} \quad y = -x \]
The slopes of these lines are \( 1 \) and \( -1 \), respectively.

5. **Setting conditions for angle bisectors**:
For \( L_1 \) to be an angle bisector, its slope \( m \) must equal \( 1 \) or \( -1 \):
- If \( m = 1 \), then \( L_1: y = x \).
- If \( m = -1 \), then \( L_1: y = -x \).

6. **Conclusion**:
Therefore, the possible values of \( m \) such that one of the lines is a bisector of the angle between the lines \( xy = 0 \) are:
\[ m = 1 \quad \text{or} \quad m = -1 \]

### Final Answer:
The possible values of \( m \) are \( 1 \) and \( -1 \).

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