What does the equation `(x - a)^(2) + ( y - b) ^(2) = c^(2)`
become when it is transferred to parallel axes through
the point (a - c, b)

Answer

To solve the problem, we need to transform the given equation \((x - a)^2 + (y - b)^2 = c^2\) to a new coordinate system that is parallel to the original axes, but shifted through the point \((a - c, b)\).

### Step-by-Step Solution:

1. **Identify the Original Equation**:
The original equation is given as:
\[ (x - a)^2 + (y - b)^2 = c^2 \]

2. **Define the New Axes**:
We are transferring to new axes that are parallel to the original axes and pass through the point \((a - c, b)\).
Let the new coordinates be \((x', y')\).

3. **Establish the Transformation**:
The transformation from the old coordinates \((x, y)\) to the new coordinates \((x', y')\) can be expressed as:
\[ x = x' + (a - c) \]
\[ y = y' + b \]

4. **Substitute the Transformation into the Original Equation**:
We substitute \(x\) and \(y\) in the original equation:
\[ (x' + (a - c) - a)^2 + (y' + b - b)^2 = c^2 \]
Simplifying this gives:
\[ (x' - c)^2 + (y')^2 = c^2 \]

5. **Expand the Equation**:
Now, we expand the left side:
\[ (x' - c)^2 + y'^2 = c^2 \]
Expanding \((x' - c)^2\):
\[ x'^2 - 2cx' + c^2 + y'^2 = c^2 \]

6. **Simplify the Equation**:
Now, we can simplify by subtracting \(c^2\) from both sides:
\[ x'^2 - 2cx' + c^2 + y'^2 - c^2 = 0 \]
This simplifies to:
\[ x'^2 + y'^2 - 2cx' = 0 \]

7. **Final Form**:
Rearranging gives us the final form of the equation:
\[ x'^2 + y'^2 = 2cx' \]

### Final Result:
The equation \((x - a)^2 + (y - b)^2 = c^2\) becomes:
\[ x'^2 + y'^2 = 2cx' \]
when transferred to parallel axes through the point \((a - c, b)\).