If `ax^(2)+2hxy+by^(2)+2gx+2fy+c=0` represents parallel straight lines, then

Options

A. `hf=bg`
B. `h^(2)=bc`
C. `a^(2)f=b^(2)g`
D. none of these

Explanation

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

### Step 1: Identify the Conditions for Parallel Lines
For the given quadratic equation in two variables to represent parallel straight lines, there are two conditions that must be satisfied:
1. \( h^2 = ab \)
2. \( bg^2 = af^2 \)

### Step 2: Rewrite the Second Condition
To analyze the second condition, we can multiply both sides of the equation \( bg^2 = af^2 \) by \( b \):
\[ b^2g^2 = abf^2 \]

### Step 3: Substitute the First Condition into the Second Condition
From the first condition \( h^2 = ab \), we can substitute \( ab \) in the second condition:
\[ b^2g^2 = h^2f^2 \]

### Step 4: Rearranging the Equation
Now we can rearrange the equation:
\[ b^2g^2 - h^2f^2 = 0 \]

### Step 5: Factor the Equation
This can be factored as:
\[ (bg - hf)(bg + hf) = 0 \]

### Step 6: Set Each Factor to Zero
Setting each factor to zero gives us two cases:
1. \( bg - hf = 0 \) which simplifies to \( bg = hf \)
2. \( bg + hf = 0 \) which is not relevant for parallel lines.

Thus, the relevant condition for the equation to represent parallel straight lines is:
\[ hf = bg \]

### Conclusion
Therefore, the condition for the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) to represent parallel straight lines is:
\[ hf = bg \]

---