The lines `y = mx` bisects the angle between the lines `ax^(2) +2hxy +by^(2) = 0` if

Options

A. `h(1+m^(2)) = m(a+b)`
B. `h(1-m^(2))=m(a-b)`
C. `h(1+m^(2))=m(a-b)`
D. None of these

Explanation

To solve the problem, we need to find the condition under which the line \( y = mx \) bisects the angle between the lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines. We can express these lines in the form \( y = m_1x \) and \( y = m_2x \), where \( m_1 \) and \( m_2 \) are the slopes of the two lines.

2. **Using the Angle Bisector Condition**:
The line \( y = mx \) bisects the angle between the two lines if it satisfies the angle bisector condition:
\[ \frac{m_1 - m}{1 + m_1m} = \frac{m - m_2}{1 + mm_2} \]
This condition can be rearranged to find a relationship between \( m, m_1, \) and \( m_2 \).

3. **Finding the Slopes**:
The slopes \( m_1 \) and \( m_2 \) can be derived from the quadratic equation formed by the given equation. The slopes are given by:
\[ m_1, m_2 = \frac{-h \pm \sqrt{h^2 - ab}}{a} \]

4. **Substituting Slopes into the Angle Bisector Condition**:
Substitute \( m_1 \) and \( m_2 \) into the angle bisector condition:
\[ \frac{\frac{-h + \sqrt{h^2 - ab}}{a} - m}{1 + \frac{-h + \sqrt{h^2 - ab}}{a}m} = \frac{m - \frac{-h - \sqrt{h^2 - ab}}{a}}{1 + m\left(\frac{-h - \sqrt{h^2 - ab}}{a}\right)} \]

5. **Cross Multiplying and Simplifying**:
After substituting the slopes, cross-multiply and simplify the resulting equation. This will lead to an equation involving \( a, b, h, \) and \( m \).

6. **Final Relation**:
After simplification, we arrive at the required relation:
\[ h(1 - m^2) = (a - b)m \]

### Final Answer:
The lines \( y = mx \) bisect the angle between the lines represented by \( ax^2 + 2hxy + by^2 = 0 \) if:
\[ h(1 - m^2) = (a - b)m \]