The ratio of the slopes of the lines represented by `ax^(2)+2hxy+by^(2)=0` is `2:3`, then `h^(2):ab=`

Options

A. `6:5`
B. `5:6`
C. `24:25`
D. `25:24`

Answer

To solve the problem, we need to find the ratio \( h^2 : ab \) given that the ratio of the slopes of the lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \) is \( 2:3 \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines through the origin. We can express \( y \) in terms of \( x \) by substituting \( y = mx \) (where \( m \) is the slope).

2. **Substituting for y**:
Substitute \( y = mx \) into the equation:
\[ ax^2 + 2h(mx)x + b(mx)^2 = 0 \]
This simplifies to:
\[ ax^2 + 2hmx^2 + bmx^2 = 0 \]
Factoring out \( x^2 \) (assuming \( x \neq 0 \)):
\[ (a + 2hm + bm^2) = 0 \]

3. **Forming the Quadratic Equation**:
Rearranging gives us a quadratic equation in \( m \):
\[ bm^2 + 2hm + a = 0 \]

4. **Finding the Slopes**:
The slopes \( m_1 \) and \( m_2 \) can be found using the quadratic formula:
\[ m = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} = \frac{-2h \pm \sqrt{(2h)^2 - 4ab}}{2b} \]
The slopes are:
\[ m_1, m_2 = \frac{-2h \pm \sqrt{4h^2 - 4ab}}{2b} \]

5. **Using the Ratio of Slopes**:
Given that the ratio of the slopes \( \frac{m_1}{m_2} = \frac{2}{3} \), we can express \( m_1 \) in terms of \( m_2 \):
\[ m_1 = \frac{2}{3} m_2 \]

6. **Setting Up the Equations**:
From the sum of the roots:
\[ m_1 + m_2 = -\frac{2h}{b} \]
Substituting \( m_1 = \frac{2}{3} m_2 \):
\[ \frac{2}{3} m_2 + m_2 = -\frac{2h}{b} \]
This simplifies to:
\[ \frac{5}{3} m_2 = -\frac{2h}{b} \implies m_2 = -\frac{6h}{5b} \]

7. **Finding m1**:
Using \( m_1 = \frac{2}{3} m_2 \):
\[ m_1 = \frac{2}{3} \left(-\frac{6h}{5b}\right) = -\frac{4h}{5b} \]

8. **Product of the Roots**:
The product of the roots is given by:
\[ m_1 m_2 = \frac{a}{b} \]
Substituting the values of \( m_1 \) and \( m_2 \):
\[ \left(-\frac{4h}{5b}\right) \left(-\frac{6h}{5b}\right) = \frac{a}{b} \]
This simplifies to:
\[ \frac{24h^2}{25b^2} = \frac{a}{b} \]

9. **Finding the Ratio \( h^2 : ab \)**:
Rearranging gives:
\[ 24h^2 = 25ab \implies \frac{h^2}{ab} = \frac{25}{24} \]

### Final Answer:
Thus, the ratio \( h^2 : ab = 25 : 24 \).