If the slope of one line in the pair `ax^(2)+4xy+y^(2)=0` is three times the other, then a =

Options

A. 3
B. 1
C. `-3`
D. `-1`

Explanation

To solve the problem, we need to find the value of \( a \) given that the slopes of the lines represented by the equation \( ax^2 + 4xy + y^2 = 0 \) have a specific relationship. Let's break this down step by step.

### Step 1: Rewrite the given equation
The equation of the pair of straight lines is given as:
\[ ax^2 + 4xy + y^2 = 0 \]
We can divide the entire equation by \( x^2 \) (assuming \( x \neq 0 \)):
\[ a + \frac{4y}{x} + \frac{y^2}{x^2} = 0 \]
Let \( m = \frac{y}{x} \). Then, we can rewrite the equation as:
\[ m^2 + 4m + a = 0 \]

### Step 2: Identify the relationship between the slopes
From the problem, we know that the slope of one line \( m_1 \) is three times the slope of the other line \( m_2 \):
\[ m_1 = 3m_2 \]

### Step 3: Use the properties of the roots of the quadratic equation
For the quadratic equation \( m^2 + 4m + a = 0 \):
- The sum of the roots \( m_1 + m_2 \) is given by:
\[ m_1 + m_2 = -\frac{b}{a} = -4 \]
- The product of the roots \( m_1 m_2 \) is given by:
\[ m_1 m_2 = \frac{c}{a} = a \]

### Step 4: Substitute \( m_1 \) in terms of \( m_2 \)
Substituting \( m_1 = 3m_2 \) into the sum of the roots:
\[ 3m_2 + m_2 = -4 \implies 4m_2 = -4 \implies m_2 = -1 \]

### Step 5: Find \( m_1 \)
Now substituting \( m_2 = -1 \) back to find \( m_1 \):
\[ m_1 = 3m_2 = 3(-1) = -3 \]

### Step 6: Use the product of the roots to find \( a \)
Now, using the product of the roots:
\[ m_1 m_2 = a \implies (-3)(-1) = a \implies a = 3 \]

### Conclusion
Thus, the value of \( a \) is:
\[ \boxed{3} \]