If the equation `ax^(2)+by^(2)+2hxy+2g x+2fy+c=0` represents a circle, the condition will be

Options

A. a=b and c=0
B. f=g and h=0
C. a=b and h=0
D. f=g and c=0

Answer

To determine the conditions under which the equation \( ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \) represents a circle, we need to analyze the coefficients of the equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The general second-degree equation is given by:
\[ ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \]
Here, the coefficients are:
- Coefficient of \(x^2\) is \(a\)
- Coefficient of \(y^2\) is \(b\)
- Coefficient of \(xy\) is \(2h\)

2. **Condition for a circle**:
For the equation to represent a circle, the following conditions must be satisfied:
- The coefficient of \(x^2\) must be equal to the coefficient of \(y^2\):
\[ a = b \]
- The coefficient of \(xy\) must be zero:
\[ 2h = 0 \quad \Rightarrow \quad h = 0 \]

3. **Final Conditions**:
Therefore, the conditions for the equation to represent a circle are:
- \(a = b\)
- \(h = 0\)

### Conclusion:
The conditions that must be satisfied for the equation \( ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \) to represent a circle are:
- \(a = b\)
- \(h = 0\)