If `m_1 and m_2` are the slopes of the lines represented by `ax^(2)+2hxy+by^(2)=0`, then `m_1+m_2=`

Options

A. `(2h)/(a)`
B. `(-2h)/(a)`
C. `(2h)/(b)`
D. `(-2h)/(b)`

Answer

To solve the problem, we need to find the sum of the slopes \( m_1 \) and \( m_2 \) of the lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \).

### Step-by-Step Solution:

1. **Understand the Equation**: The given equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines through the origin.

2. **Formulate the Lines**: The lines can be expressed in the form \( y = m_1x \) and \( y = m_2x \), where \( m_1 \) and \( m_2 \) are the slopes of the lines.

3. **Set Up the Product of the Lines**: The product of the lines can be expressed as:
\[ (y - m_1x)(y - m_2x) = 0 \]
Expanding this gives:
\[ y^2 - (m_1 + m_2)xy + m_1m_2x^2 = 0 \]

4. **Compare with the Given Equation**: Now, we need to compare this equation with the given equation \( ax^2 + 2hxy + by^2 = 0 \). Rearranging the terms, we can write it as:
\[ by^2 + 2hxy + ax^2 = 0 \]

5. **Identify Coefficients**: From the expanded form \( y^2 - (m_1 + m_2)xy + m_1m_2x^2 = 0 \), we can identify:
- Coefficient of \( xy \): \( -(m_1 + m_2) \)
- Coefficient of \( x^2 \): \( m_1m_2 \)

From the equation \( ax^2 + 2hxy + by^2 = 0 \), we can identify:
- Coefficient of \( xy \): \( 2h \)
- Coefficient of \( x^2 \): \( a \)

6. **Set Up the Equation**: Since the coefficients of \( xy \) must be equal, we have:
\[ -(m_1 + m_2) = 2h \]

7. **Solve for \( m_1 + m_2 \)**: Rearranging the equation gives:
\[ m_1 + m_2 = -2h \]

8. **Final Result**: Thus, the sum of the slopes \( m_1 + m_2 \) is:
\[ m_1 + m_2 = \frac{2h}{b} \]

### Final Answer:
\[ m_1 + m_2 = \frac{2h}{b} \]