If the slope of one of the lines represented by `ax^(2)+2hxy+by^(2)=0` is the square of the other , then `(a+b)/(h)+(8h^(2))/(ab)=`

Options

A. 4
B. 6
C. 8
D. None of these

Explanation

To solve the problem, we need to find the value of \((a+b)/h + (8h^2)/(ab)\) given that one slope of the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\) is the square of the other.

### Step-by-Step Solution:

1. **Identify the slopes**:
Let the slopes of the lines be \(m_1 = m\) and \(m_2 = m^2\).

2. **Use the relationships for the slopes**:
From the properties of the pair of lines represented by the equation, we know:
\[ m_1 + m_2 = -\frac{2h}{b} \quad \text{(1)} \]
\[ m_1 m_2 = \frac{a}{b} \quad \text{(2)} \]

3. **Substitute the slopes**:
Substitute \(m_1\) and \(m_2\) into equations (1) and (2):
\[ m + m^2 = -\frac{2h}{b} \quad \text{(3)} \]
\[ m \cdot m^2 = m^3 = \frac{a}{b} \quad \text{(4)} \]

4. **Express \(m^2\) in terms of \(m\)**:
From equation (3), we can rearrange it:
\[ m^2 + m + \frac{2h}{b} = 0 \]

5. **Use the cubic relationship**:
Now, we can express \(m^3\) using equation (4):
\[ m^3 = \frac{a}{b} \]

6. **Substitute \(m^3\) into the cubic equation**:
From \(m + m^2 = -\frac{2h}{b}\), we can cube both sides:
\[ (m + m^2)^3 = \left(-\frac{2h}{b}\right)^3 \]
Expanding the left side gives:
\[ m^3 + m^6 + 3m^2m(-\frac{2h}{b}) = -\frac{8h^3}{b^3} \]

7. **Substituting for \(m^3\)**:
Substitute \(m^3 = \frac{a}{b}\) into the equation:
\[ \frac{a}{b} + \left(\frac{a}{b}\right)^2 + 3m^2(-\frac{2h}{b}) = -\frac{8h^3}{b^3} \]

8. **Rearranging and simplifying**:
Multiply through by \(b^2\) to eliminate the denominators:
\[ ab + a^2 - 6hm^2 = -8h^3 \]

9. **Factor out common terms**:
Rearranging gives:
\[ ab + a^2 + 8h^3 = 6hm^2 \]

10. **Final expression**:
Dividing through by \(h\):
\[ \frac{a+b}{h} + \frac{8h^2}{ab} = 6 \]

Thus, the final answer is:
\[ \frac{a+b}{h} + \frac{8h^2}{ab} = 6 \]