The equation `ax^(2)+2sqrt(ab)xy+by^(2)+2 gx+2fy+c=0` represents a pair of parallel straight lines, if

Options

A. `ag=bf^(2)`
B. `a^(2)g=b^(2)f`
C. `bg^(2)=af^(2)`
D. `b^(2)g=a^(2)f`

Explanation

To determine the condition under which the equation

\[ ax^2 + 2\sqrt{ab}xy + by^2 + 2gx + 2fy + c = 0 \]

represents a pair of parallel straight lines, we can follow these steps:

### Step 1: Identify the coefficients
The given equation can be compared with the general second-degree equation:

\[ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \]

From the given equation, we can identify:
- \( A = a \)
- \( H = \sqrt{ab} \)
- \( B = b \)
- \( G = g \)
- \( F = f \)
- \( C = c \)

### Step 2: Condition for parallel lines
For the equation to represent a pair of parallel straight lines, the following condition must hold:

\[ \frac{A}{H} = \frac{H}{B} = \frac{G}{F} \]

### Step 3: Substitute the values
Substituting the identified values into the condition:

1. From \( \frac{A}{H} = \frac{H}{B} \):
\[ \frac{a}{\sqrt{ab}} = \frac{\sqrt{ab}}{b} \]

2. Cross-multiplying gives:
\[ a \cdot b = ab \quad \text{(This is always true)} \]

### Step 4: Consider the second part of the condition
Now we consider the second part of the condition \( \frac{H}{B} = \frac{G}{F} \):

\[ \frac{\sqrt{ab}}{b} = \frac{g}{f} \]

Cross-multiplying gives:

\[ \sqrt{ab} \cdot f = b \cdot g \]

### Step 5: Square both sides
Squaring both sides of the equation:

\[ abf^2 = b^2g^2 \]

### Step 6: Rearranging
Rearranging gives us the final condition:

\[ af^2 = bg^2 \]

### Conclusion
Thus, the equation represents a pair of parallel straight lines if:

\[ af^2 = bg^2 \]