If the pair of lines represented by `ax^(2)+2hxy+by^(2)=0, b ne 0`, are such that the sum of the slopes of the lines is three times the product of their slopes, then

Options

A. `3b+2h=0`
B. `3a+2h=0`
C. `2a+3h=0`
D. none of these

Explanation

To solve the problem, we need to analyze the given equation of the pair of lines and derive the relationship between the coefficients based on the condition provided.

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ ax^2 + 2hxy + by^2 = 0 \]
where \( b \neq 0 \).

2. **Divide the entire equation by \( x^2 \)**:
\[ \frac{a}{x^2} x^2 + \frac{2h}{x} y + \frac{b}{x^2} y^2 = 0 \implies b \left(\frac{y}{x}\right)^2 + 2h \left(\frac{y}{x}\right) + a = 0 \]
Let \( m = \frac{y}{x} \) (the slope of the lines).

3. **Rewrite the equation in terms of \( m \)**:
\[ bm^2 + 2hm + a = 0 \]
This is a quadratic equation in \( m \).

4. **Identify the sum and product of the roots**:
- The sum of the slopes \( m_1 + m_2 = -\frac{b}{a} = -\frac{2h}{b} \)
- The product of the slopes \( m_1 m_2 = \frac{c}{a} = \frac{a}{b} \)

5. **Use the given condition**:
We are given that the sum of the slopes is three times the product of the slopes:
\[ m_1 + m_2 = 3(m_1 m_2) \]
Substituting the expressions we found:
\[ -\frac{2h}{b} = 3 \left(\frac{a}{b}\right) \]

6. **Multiply through by \( b \) to eliminate the denominator**:
\[ -2h = 3a \]

7. **Rearranging gives**:
\[ 3a + 2h = 0 \]

### Conclusion:
The relationship derived from the given condition is:
\[ 3a + 2h = 0 \]