If a circle of radius 2 touches X-axis at (1,0) then its centre may be

Options

A. (1,2) an (1,-2)
B. (1,2) and (2,1)
C. (-1,2)(1,-2)
D. (-1,2)(-1,-2)

Answer

To find the possible centers of a circle with a radius of 2 that touches the X-axis at the point (1, 0), we can follow these steps:

### Step 1: Understand the Geometry
The circle touches the X-axis at the point (1, 0). This means that the distance from the center of the circle to the X-axis is equal to the radius of the circle.

### Step 2: Determine the Y-coordinate of the Center
Since the radius of the circle is 2, the center of the circle must be 2 units away from the X-axis. Therefore, the Y-coordinate of the center can either be 2 units above or 2 units below the X-axis.

### Step 3: Calculate the Possible Centers
1. **Above the X-axis**: If the center is above the X-axis, the Y-coordinate will be:
\[ Y = 0 + 2 = 2 \]
Thus, the center will be at the point (1, 2).

2. **Below the X-axis**: If the center is below the X-axis, the Y-coordinate will be:
\[ Y = 0 - 2 = -2 \]
Thus, the center will be at the point (1, -2).

### Step 4: Conclusion
The possible centers of the circle are:
- (1, 2) (above the X-axis)
- (1, -2) (below the X-axis)

### Final Answer
The centers of the circle may be (1, 2) and (1, -2).

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