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The problem describes a circle with center \( C \), radius \( r \), and it touches the x-axis at the point \( (12, 0) \). We are asked to find the equation of the circle.

### Step-by-step solution:

1. **Identify the center coordinates:**
   - The circle touches the x-axis, which implies that the vertical distance from the center \( C \) to the x-axis is equal to the radius \( r \).
   - The point of contact with the x-axis is at \( (12, 0) \), meaning that the distance from the center \( C \) to this point is the radius \( r \). Therefore, the x-coordinate of the center \( C \) is \( 12 \), and the y-coordinate must be \( r \) (since the circle is above the x-axis).

   Thus, the center \( C \) has coordinates \( (12, r) \).

2. **Form the equation of the circle:**
   - The general equation of a circle with center \( (h, k) \) and radius \( r \) is:
     \[
     (x - h)^2 + (y - k)^2 = r^2
     \]
   - Substituting \( h = 12 \) and \( k = r \), the equation becomes:
     \[
     (x - 12)^2 + (y - r)^2 = r^2
     \]

### Final Answer:
The equation of the circle is \( (x - 12)^2 + (y - r)^2 = r^2 \).

If you have any questions or need further details, feel free to ask!

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### Related questions:
1. What is the significance of the circle touching the x-axis?
2. How do you find the radius of the circle given the center and a point on the circumference?
3. What would be the equation of the circle if it touched the y-axis instead?
4. How can you determine if a point lies inside, on, or outside the circle?
5. What are the properties of a circle in a coordinate plane?

### Tip:
Remember that when a circle touches a coordinate axis, the radius is always equal to the perpendicular distance from the center to that axis.

The diagram shows a circle with center C. The circle has radius r and touches the x-axis at (12, 0). Work out the equation of the circle.

Find the equation of a circle with center C and radius r that touches the x-axis at (12, 0). The solution involves key geometry concepts and the standard equation of a circle. Suitable for students in Grades 10-12.

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