If `h^(2)=ab` then slopes of lines `ax^(2)+2hxy+by^(2)=0` are in the ratio

Options

A. `1:2`
B. `2:1`
C. `2:3`
D. `1:1`

Answer

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

### Step-by-Step Solution:

1. **Identify the Equation**:
The given equation is \( ax^2 + 2hxy + by^2 = 0 \). This can be treated as a quadratic equation in \( y \).

2. **Rewrite the Equation**:
We can rearrange the equation in the standard form of a quadratic:
\[ by^2 + 2hxy + ax^2 = 0 \]
Here, we can identify \( A = b \), \( B = 2hx \), and \( C = ax^2 \).

3. **Use the Quadratic Formula**:
The roots of the quadratic equation \( Ay^2 + By + C = 0 \) can be found using the quadratic formula:
\[ y = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
In our case, the slopes \( m_1 \) and \( m_2 \) of the lines can be expressed as:
\[ m_1, m_2 = \frac{-2hx \pm \sqrt{(2hx)^2 - 4b(ax^2)}}{2b} \]

4. **Calculate the Discriminant**:
The discriminant \( D \) is given by:
\[ D = (2hx)^2 - 4b(ax^2) = 4h^2x^2 - 4abx^2 \]
Since \( h^2 = ab \), we substitute this into the discriminant:
\[ D = 4abx^2 - 4abx^2 = 0 \]
This indicates that the roots are equal.

5. **Find the Sum and Product of the Roots**:
The sum of the roots \( m_1 + m_2 \) and the product of the roots \( m_1 m_2 \) can be calculated as:
\[ m_1 + m_2 = -\frac{B}{A} = -\frac{2h}{b} \]
\[ m_1 m_2 = \frac{C}{A} = \frac{a}{b} \]

6. **Use the Relationship Between the Roots**:
Since the roots are equal, we can set \( m_1 = m_2 = m \). Thus:
\[ m + m = -\frac{2h}{b} \implies 2m = -\frac{2h}{b} \implies m = -\frac{h}{b} \]

7. **Find the Ratio of the Slopes**:
Since \( m_1 = m_2 \), the ratio \( \frac{m_1}{m_2} = 1 \). Therefore, the ratio of the slopes of the lines is:
\[ m_1 : m_2 = 1 : 1 \]

### Final Answer:
The slopes of the lines are in the ratio \( 1 : 1 \).