The equation `x^(2)+y^(2)+2gx+c=0` where g is a parameter and c is a constant represents a family of coaxial circles any two members of which have the radical axis

Options

A. x-axis
B. y-axis
C. x=const.
D. y=const.

Answer

To solve the problem, we need to show that the given equation represents a family of coaxial circles and that any two members of this family have a radical axis.

### Step-by-Step Solution:

1. **Identify the given equation**:
The equation provided is:
\[ x^2 + y^2 + 2gx + c = 0 \]
where \( g \) is a parameter and \( c \) is a constant.

2. **Rearranging the equation**:
We can rewrite the equation in the standard form of a circle:
\[ x^2 + y^2 + 2gx + c = 0 \implies x^2 + y^2 + 2gx = -c \]
This can be further rearranged to:
\[ (x + g)^2 + y^2 = g^2 - c \]
This shows that the center of the circle is at \((-g, 0)\) and the radius is \(\sqrt{g^2 - c}\).

3. **Understanding coaxial circles**:
A family of circles is coaxial if they share the same radical axis. In our case, since \( g \) varies and \( c \) is constant, the centers of the circles will lie on the line \( y = 0 \) (the x-axis) and will be at different positions along this line.

4. **Finding the radical axis**:
Let’s consider two circles from this family:
- Circle 1: \( S_1: x^2 + y^2 + 2g_1x + c = 0 \)
- Circle 2: \( S_2: x^2 + y^2 + 2g_2x + c = 0 \)

The radical axis can be found using the formula:
\[ S_1 - S_2 = 0 \]
Substituting the equations of the circles:
\[ (x^2 + y^2 + 2g_1x + c) - (x^2 + y^2 + 2g_2x + c) = 0 \]
Simplifying this gives:
\[ 2g_1x - 2g_2x = 0 \implies 2(g_1 - g_2)x = 0 \]
This implies:
\[ x(g_1 - g_2) = 0 \]
Since \( g_1 \) and \( g_2 \) are different (as they represent different circles), we conclude that:
\[ x = 0 \]
This means the radical axis is the line \( x = 0 \), which is the y-axis.

5. **Conclusion**:
Therefore, we have shown that the given equation represents a family of coaxial circles, and any two members of this family have the radical axis along the y-axis (i.e., \( x = 0 \)).