Two equal circles with their centres as x and y axis will possess the radical axis in the following form

Options

A. `ax-by-(a^(2)+b^(2))/(4)=0`
B. `2gx-2fy+g^(2)-f^(2)=0`
C. `g^(2)x+f^(2)y-g^(4)-f^(4)=0`
D. `2g^(2)x+2f^(2)y-g^(4)-f^(4)=0`

Explanation

To find the radical axis of two equal circles centered on the x-axis and y-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Circles**:
- Let the first circle \( S_1 \) have its center at \( (a, 0) \) on the x-axis and radius \( r \).
- The equation of the first circle \( S_1 \) is:
\[ (x - a)^2 + y^2 = r^2 \]

2. **Define the Second Circle**:
- Let the second circle \( S_2 \) have its center at \( (0, b) \) on the y-axis and radius \( r \).
- The equation of the second circle \( S_2 \) is:
\[ x^2 + (y - b)^2 = r^2 \]

3. **Set Up the Radical Axis**:
- The radical axis is given by the equation \( S_1 - S_2 = 0 \).
- Thus, we have:
\[ \left((x - a)^2 + y^2 - r^2\right) - \left(x^2 + (y - b)^2 - r^2\right) = 0 \]

4. **Expand the Equations**:
- Expanding \( S_1 \):
\[ (x - a)^2 + y^2 = x^2 - 2ax + a^2 + y^2 \]
- Expanding \( S_2 \):
\[ x^2 + (y - b)^2 = x^2 + y^2 - 2by + b^2 \]

5. **Combine and Simplify**:
- Substitute the expansions into the radical axis equation:
\[ (x^2 - 2ax + a^2 + y^2 - r^2) - (x^2 + y^2 - 2by + b^2 - r^2) = 0 \]
- This simplifies to:
\[ -2ax + a^2 + 2by - b^2 = 0 \]
- Rearranging gives:
\[ 2by - 2ax + (a^2 - b^2) = 0 \]

6. **Final Form**:
- The equation can be rewritten as:
\[ 2ax - 2by + (b^2 - a^2) = 0 \]
- This is in the form \( 2gx - 2fy + (g^2 - f^2) = 0 \).

### Conclusion:
Thus, the radical axis of the two equal circles centered on the x-axis and y-axis can be expressed as:
\[ 2ax - 2by + (b^2 - a^2) = 0 \]