If the equation `ax^(2)+2hxy+by^(2)+2gx+2fy+c=0` represents a pair of parallel lines, then

Options

A. `a/h=b/h=f/g`
B. `a/h=h/b=f/g`
C. `a/h=h/b=g/f`
D. none of these

Answer

To determine the conditions under which the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) represents a pair of parallel lines, we can follow these steps:

### Step 1: Understand the Form of the Equation
The given equation is a general second-degree equation in two variables \( x \) and \( y \). For it to represent a pair of parallel lines, it must be factored into the form \( (l_1)(l_2) = 0 \), where \( l_1 \) and \( l_2 \) are linear equations.

### Step 2: Identify the Conditions for Parallel Lines
For the lines represented by the equation to be parallel, the coefficients of \( x \) and \( y \) in the linear factors must be proportional. This means that the slopes of the lines must be equal.

### Step 3: Set Up the Conditions
Assuming the lines can be expressed as:
- \( l_1: a_1 x + b_1 y + c_1 = 0 \)
- \( l_2: a_2 x + b_2 y + c_2 = 0 \)

The condition for parallel lines is:
\[ \frac{a_1}{b_1} = \frac{a_2}{b_2} \]

### Step 4: Relate Coefficients to the Original Equation
From the original equation, we can relate:
- \( a_1 = a \)
- \( b_1 = b \)
- \( h = \frac{1}{2}(a_1 b_2 + a_2 b_1) \)

### Step 5: Use the Discriminant Condition
For the equation to represent a pair of lines, the discriminant must be zero. The condition for the discriminant \( D \) of the quadratic form to equal zero is:
\[ D = h^2 - ab = 0 \]
This implies:
\[ h^2 = ab \]

### Step 6: Conclude the Conditions
Thus, for the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) to represent a pair of parallel lines, the following conditions must hold:
1. \( h^2 = ab \)
2. The lines must be parallel, which can be expressed as \( \frac{a}{b} = \frac{h}{g} \).

### Final Answer
The equation represents a pair of parallel lines if \( h^2 = ab \).

---