If `g^(2)+f^(2)=c`, then the equation `x^(2)+y^(2)+2gx+2fy+c=0` will represent

Options

A. a circle of radius g
B. a circle of radius f
C. a circle of diameter `sqrt(c)`
D. a circle of radius 0

Explanation

To solve the problem, we need to analyze the given equation and apply the conditions provided.

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

2. **Rearrange the equation**:
We can rearrange the equation to isolate the constant term on one side:
\[ x^2 + y^2 + 2gx + 2fy = -c \]

3. **Complete the square**:
We will complete the square for both \(x\) and \(y\):
- For \(x\):
\[ x^2 + 2gx = (x + g)^2 - g^2 \]
- For \(y\):
\[ y^2 + 2fy = (y + f)^2 - f^2 \]
- Substitute these back into the equation:
\[ (x + g)^2 - g^2 + (y + f)^2 - f^2 = -c \]
- Simplifying gives:
\[ (x + g)^2 + (y + f)^2 = g^2 + f^2 - c \]

4. **Use the given condition**:
We know from the problem statement that \(g^2 + f^2 = c\). Substituting this into our equation:
\[ (x + g)^2 + (y + f)^2 = c - c = 0 \]

5. **Interpret the result**:
The equation \((x + g)^2 + (y + f)^2 = 0\) implies that both squares must equal zero. Therefore:
\[ x + g = 0 \quad \text{and} \quad y + f = 0 \]
This means:
\[ x = -g \quad \text{and} \quad y = -f \]
Hence, the equation represents a single point, which is the center of the circle.

6. **Conclusion**:
Since the radius is zero, the equation represents a point circle. Therefore, the correct answer is that the equation represents a circle of radius 0.

### Final Answer:
The equation \(x^2 + y^2 + 2gx + 2fy + c = 0\) represents a circle of radius 0.

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