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

Options

A. 0
B. `pi//2`
C. `pi`
D. `3pi//2`

Answer

To solve the problem, we need to determine the value of \( \cos^{-1}(m) \) given 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 \( xy = 0 \).

### Step-by-Step Solution:

1. **Identify the lines represented by \( xy = 0 \)**:
The equation \( xy = 0 \) represents two lines: \( x = 0 \) (the y-axis) and \( y = 0 \) (the x-axis). The angle between these two lines is \( 90^\circ \).

**Hint**: Remember that the lines \( x = 0 \) and \( y = 0 \) are perpendicular to each other.

2. **Determine the slopes of the angle bisectors**:
The angle bisectors of the lines \( x = 0 \) and \( y = 0 \) can be found. Since these lines are perpendicular, the slopes of the angle bisectors will be \( 1 \) and \( -1 \).

**Hint**: The slopes of the angle bisectors of two perpendicular lines can be derived from the angles they form.

3. **Rewrite the given equation**:
The equation \( my^2 + (1 - m^2)xy - mx^2 = 0 \) can be factored to find the individual lines. We can rearrange and factor it as follows:
\[ my^2 + (1 - m^2)xy - mx^2 = 0 \]
Rearranging gives:
\[ m y^2 + (1 - m^2)xy - mx^2 = 0 \]

4. **Factor the quadratic**:
Factoring the quadratic equation, we can express it as:
\[ (y - mx)(y + \frac{x}{m}) = 0 \]
This gives us the two lines:
\[ y = mx \quad \text{and} \quad y = -\frac{1}{m}x \]

**Hint**: Factoring a quadratic can help identify the slopes of the lines represented.

5. **Set the slopes equal to the bisector slopes**:
We know that one of these lines must be a bisector of the angle between the x-axis and y-axis. Therefore, we set:
\[ m = 1 \quad \text{or} \quad -\frac{1}{m} = -1 \]
From \( m = 1 \), we have one solution. From \( -\frac{1}{m} = -1 \), we find \( m = 1 \).

**Hint**: The slopes of the lines must match the slopes of the angle bisectors.

6. **Calculate \( \cos^{-1}(m) \)**:
If \( m = 1 \), then:
\[ \cos^{-1}(m) = \cos^{-1}(1) = 0^\circ \]
If \( m = -1 \), then:
\[ \cos^{-1}(m) = \cos^{-1}(-1) = 180^\circ \]

**Hint**: The cosine inverse function gives angles based on the value of \( m \).

7. **Final Answer**:
Therefore, the values of \( \cos^{-1}(m) \) can be \( 0^\circ \) or \( 180^\circ \).

### Summary:
The final answer is that \( \cos^{-1}(m) \) can take values of either \( 0^\circ \) or \( 180^\circ \) based on the values of \( m \) derived from the angle bisector condition.