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

Options

A. `sqrt((g^(2)-ac)/(h^(2)+a^(2)))`
B. `2sqrt((g^(2)-ac)/(a(a+b)))`
C. `sqrt((g^(2)+ac)/(a(a+b)))`
D. `2sqrt((g^(2)+ac)/(a(a+b)))`

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 General Form
The equation given is a second-degree equation in two variables \( x \) and \( y \). For it to represent a pair of parallel lines, the discriminant of the quadratic must be zero.

### Step 2: Identify the Coefficients
From the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \), we identify:
- \( A = a \)
- \( B = 2h \)
- \( C = b \)
- \( D = 2g \)
- \( E = 2f \)
- \( F = c \)

### Step 3: Condition for Parallel Lines
For the equation to represent a pair of parallel lines, the following condition must hold:
\[ \text{Condition: } B^2 - 4AC = 0 \]
This means:
\[ (2h)^2 - 4ab = 0 \]
Simplifying gives:
\[ 4h^2 - 4ab = 0 \]
Thus, we can simplify this to:
\[ h^2 = ab \]

### Step 4: Conclusion
The condition for the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) to represent a pair of parallel lines is:
\[ h^2 = ab \]

### Summary
If \( h^2 = ab \), then the given equation represents a pair of parallel lines.

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