If the slopes of the lines given by `ax^(2)+2hxy+by^(2)=0` are in the ratio `3:1`, then `h^(2)=`

Options

A. `(ab)/(3)`
B. `(4ab)/(3)`
C. `(4a)/(3b)`
D. none of these

Explanation

To solve the problem, we start with the equation of the pair of straight lines given by:

\[ ax^2 + 2hxy + by^2 = 0 \]

We need to find \( h^2 \) given that the slopes of the lines are in the ratio \( 3:1 \).

### Step 1: Rewrite the equation
We can rewrite the equation by dividing through by \( x^2 \):

\[ \frac{a}{x^2} + \frac{2h}{x} \cdot \frac{y}{x} + \frac{b}{x^2} \cdot \left(\frac{y}{x}\right)^2 = 0 \]

Let \( m = \frac{y}{x} \). Then the equation becomes:

\[ am^2 + 2hm + b = 0 \]

### Step 2: Identify the slopes
The roots of this quadratic equation \( am^2 + 2hm + b = 0 \) represent the slopes of the lines, which we denote as \( m_1 \) and \( m_2 \).

### Step 3: Use the relationship of slopes
We are given that the slopes are in the ratio \( 3:1 \). Therefore, we can express this relationship as:

\[ m_1 = 3m_2 \]

### Step 4: Sum of the slopes
From the properties of quadratic equations, the sum of the roots (slopes) is given by:

\[ m_1 + m_2 = -\frac{2h}{a} \]

Substituting \( m_1 = 3m_2 \) into the equation:

\[ 3m_2 + m_2 = -\frac{2h}{a} \]

This simplifies to:

\[ 4m_2 = -\frac{2h}{a} \]

### Step 5: Solve for \( m_2 \)
From the above equation, we can solve for \( m_2 \):

\[ m_2 = -\frac{h}{2a} \]

### Step 6: Product of the slopes
The product of the slopes \( m_1 \cdot m_2 \) is given by:

\[ m_1 m_2 = \frac{b}{a} \]

Substituting \( m_1 = 3m_2 \):

\[ 3m_2 \cdot m_2 = \frac{b}{a} \]

This leads to:

\[ 3m_2^2 = \frac{b}{a} \]

### Step 7: Substitute \( m_2 \)
Now substituting \( m_2 = -\frac{h}{2a} \) into the equation:

\[ 3\left(-\frac{h}{2a}\right)^2 = \frac{b}{a} \]

This simplifies to:

\[ 3 \cdot \frac{h^2}{4a^2} = \frac{b}{a} \]

### Step 8: Rearranging the equation
Multiplying both sides by \( 4a^2 \):

\[ 3h^2 = 4ab \]

### Step 9: Solve for \( h^2 \)
Finally, we can solve for \( h^2 \):

\[ h^2 = \frac{4ab}{3} \]

### Final Answer
Thus, the value of \( h^2 \) is:

\[ \boxed{\frac{4ab}{3}} \]