The circle `(x-r) ^(2) + (y-r) ^(2) =r ^(2)` touches

Options

A. x-axis at the origin
B. y-axis at the origin
C. both the coordinate axes
D. none of these

Answer

To solve the problem, we need to analyze the given circle equation and determine which axes it touches.

### Step-by-step Solution:

1. **Identify the Circle's Equation**:
The equation of the circle is given as:
\[ (x - r)^2 + (y - r)^2 = r^2 \]

2. **Determine the Center and Radius**:
From the standard form of the circle's equation \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- Center \((h, k) = (r, r)\)
- Radius \(r\)

3. **Understanding the Position of the Circle**:
The center of the circle is at the point \((r, r)\) in the Cartesian coordinate system. The radius of the circle is \(r\).

4. **Calculate the Distance from the Center to the Axes**:
- The distance from the center \((r, r)\) to the x-axis (y = 0) is \(r\).
- The distance from the center \((r, r)\) to the y-axis (x = 0) is also \(r\).

5. **Determine if the Circle Touches the Axes**:
Since the radius of the circle is equal to the distance from the center to both axes, the circle will touch both the x-axis and the y-axis.

6. **Conclusion**:
Therefore, the circle touches both the x-axis and the y-axis.

### Final Answer:
The circle touches both the x-axis and the y-axis.

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