For the equation
`ax^(2) +by^(2) + 2hxy + 2gx + 2fy + c =0` where `a ne 0`, to represent 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 Standard Form of a Circle**:
The standard form of a circle's equation is

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

Here, the coefficients of \(x^2\) and \(y^2\) are both equal to 1.

2. **Compare Coefficients**:
In the given equation, the coefficients of \(x^2\) and \(y^2\) are \(a\) and \(b\) respectively. For the equation to represent a circle, these coefficients must be equal:

\[ a = b \]

3. **Condition on the Mixed Term**:
The equation of a circle does not have any mixed term (xy term). In the given equation, the coefficient of the \(xy\) term is \(2h\). For the equation to represent a circle, this coefficient must be zero:

\[ 2h = 0 \]
\[ h = 0 \]

4. **Summary of Conditions**:
Therefore, for the equation

\[ ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \]

to represent a circle, the following conditions must be satisfied:
- \( a = b \)
- \( h = 0 \)

### Final Answer:
The conditions for the equation to represent a circle are:
- \( a = b \)
- \( h = 0 \)