If the slopes of the lines `3x^(2) + 2hxy + 4y^(2)=0` are in the ration `3:1`, then h equals

Options

A. `1//4`
B. -4
C. 0
D. `-1//4`

Answer

To find the value of \( h \) such that the slopes of the lines given by the equation \( 3x^2 + 2hxy + 4y^2 = 0 \) are in the ratio \( 3:1 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation is of the form \( ax^2 + 2hxy + by^2 = 0 \).
Here, we have:
- \( a = 3 \)
- \( b = 4 \)
- \( 2h = 2h \)

2. **Use the formulas for the sum and product of slopes**:
For the general conic equation \( ax^2 + 2hxy + by^2 = 0 \):
- The sum of the slopes \( m_1 + m_2 \) is given by:
\[ m_1 + m_2 = -\frac{2h}{b} \]
- The product of the slopes \( m_1 m_2 \) is given by:
\[ m_1 m_2 = \frac{a}{b} \]

3. **Substitute the values of \( a \) and \( b \)**:
- From the product of slopes:
\[ m_1 m_2 = \frac{3}{4} \]
- From the sum of slopes:
\[ m_1 + m_2 = -\frac{2h}{4} = -\frac{h}{2} \]

4. **Express the slopes in terms of a single variable**:
Since the slopes are in the ratio \( 3:1 \), we can let:
\[ m_1 = 3k \quad \text{and} \quad m_2 = k \]
where \( k \) is a constant.

5. **Set up equations using the sum and product**:
- From the product:
\[ (3k)(k) = \frac{3}{4} \implies 3k^2 = \frac{3}{4} \implies k^2 = \frac{1}{4} \implies k = \frac{1}{2} \]
- Thus, we have:
\[ m_2 = k = \frac{1}{2} \quad \text{and} \quad m_1 = 3k = \frac{3}{2} \]

6. **Substituting back to find \( h \)**:
- Now substitute \( m_1 \) and \( m_2 \) into the sum equation:
\[ m_1 + m_2 = \frac{3}{2} + \frac{1}{2} = 2 \]
- Set this equal to the expression for the sum of slopes:
\[ 2 = -\frac{h}{2} \]
- Solving for \( h \):
\[ -\frac{h}{2} = 2 \implies h = -4 \]

### Final Answer:
Thus, the value of \( h \) is \( -4 \).