The circle represented by the equation `x ^(2) + y^(2) + 2gx + 2fy + c=0` will be a point circle, if

Options

A. `g ^(2) + f^(2) =c`
B. `g ^(2) + f^(2) gt c`
C. `g ^(2) + f^(2) + c=0`
D. `g ^(2) + f^(2) lt c`

Answer

To determine the condition under which the circle represented by the equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \) becomes a point circle, we follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circle Equation**:
The general equation of a circle is given by \( x^2 + y^2 + 2gx + 2fy + c = 0 \). Here, \( g \), \( f \), and \( c \) are constants.

2. **Identifying the Radius**:
The radius \( r \) of the circle can be derived from the standard form of the circle equation. The radius is given by the formula:
\[ r = \sqrt{g^2 + f^2 - c} \]

3. **Condition for Point Circle**:
A point circle is defined as a circle with a radius of 0. Therefore, we set the radius equal to 0:
\[ \sqrt{g^2 + f^2 - c} = 0 \]

4. **Squaring Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ g^2 + f^2 - c = 0 \]

5. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ g^2 + f^2 = c \]

### Conclusion:
The condition for the circle to be a point circle is:
\[ g^2 + f^2 = c \]