A circle passes through the points (2, 2) and (9, 9) and touches the x-axis. The absolute value of the difference of possible x-coordinate of the point of contact is______.

Explanation

To solve the problem, we need to find the absolute value of the difference of the possible x-coordinates of the point where the circle touches the x-axis, given that it passes through the points (2, 2) and (9, 9).

### Step-by-Step Solution:

1. **Identify the center of the circle**:
Let the center of the circle be at the point \((h, k)\). Since the circle touches the x-axis, the distance from the center to the x-axis must equal the radius \(r\). Therefore, \(k = r\).

2. **Equation of the circle**:
The general equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(k\) with \(r\), we have:
\[ (x - h)^2 + (y - r)^2 = r^2 \]

3. **Substituting the points (2, 2) and (9, 9)**:
Since the circle passes through the points (2, 2) and (9, 9), we can substitute these points into the circle's equation.

For point (2, 2):
\[ (2 - h)^2 + (2 - r)^2 = r^2 \]
Expanding this gives:
\[ (2 - h)^2 + (2 - r)^2 = r^2 \]
\[ (2 - h)^2 + (2 - r)^2 - r^2 = 0 \quad \text{(1)} \]

For point (9, 9):
\[ (9 - h)^2 + (9 - r)^2 = r^2 \]
Expanding this gives:
\[ (9 - h)^2 + (9 - r)^2 - r^2 = 0 \quad \text{(2)} \]

4. **Setting up the equations**:
From equation (1):
\[ (2 - h)^2 + (2 - r)^2 - r^2 = 0 \]
Simplifying:
\[ (2 - h)^2 + 4 - 4r + r^2 - r^2 = 0 \]
\[ (2 - h)^2 + 4 - 4r = 0 \quad \text{(3)} \]

From equation (2):
\[ (9 - h)^2 + (9 - r)^2 - r^2 = 0 \]
Simplifying:
\[ (9 - h)^2 + 81 - 18r + r^2 - r^2 = 0 \]
\[ (9 - h)^2 + 81 - 18r = 0 \quad \text{(4)} \]

5. **Solving the equations**:
Now we have two equations (3) and (4):
\[ (2 - h)^2 + 4 - 4r = 0 \]
\[ (9 - h)^2 + 81 - 18r = 0 \]

From equation (3):
\[ 4r = (2 - h)^2 + 4 \implies r = \frac{(2 - h)^2 + 4}{4} \]

Substitute \(r\) into equation (4):
\[ (9 - h)^2 + 81 - 18\left(\frac{(2 - h)^2 + 4}{4}\right) = 0 \]

Simplifying this gives:
\[ (9 - h)^2 + 81 - \frac{18((2 - h)^2 + 4)}{4} = 0 \]

After solving this equation, we find two possible values for \(h\).

6. **Finding the absolute difference**:
Let the two possible x-coordinates of the point of contact be \(h_1\) and \(h_2\). The absolute value of the difference is:
\[ |h_1 - h_2| \]

After calculating, we find:
\[ |h_1 - h_2| = 12 \]

### Final Answer:
The absolute value of the difference of possible x-coordinates of the point of contact is **12**.