The positive x-axis and the negative y-axis are tangent lines to a circle, C. Given that the line x = 12 is also a tangent line to C, find the equation of C in general form.

To find the equation of the circle \( C \) in general form, we can use the given information about the tangents to the circle.

### Step 1: Analyze the Tangent Lines
The tangents given are:
1. The positive \( x \)-axis.
2. The negative \( y \)-axis.
3. The vertical line \( x = 12 \).

These tangents imply that:
- The circle is tangent to the \( x \)-axis, so the radius \( r \) of the circle equals the \( y \)-coordinate of the center.
- The circle is tangent to the \( y \)-axis, so the radius \( r \) also equals the absolute value of the \( x \)-coordinate of the center.

Since the circle is tangent to both axes at positive and negative directions, the center must be in the first quadrant, specifically at a point \((h, k)\) where \( h = k = r \).

### Step 2: Use the Tangent Line \( x = 12 \)
The line \( x = 12 \) is a vertical tangent to the circle, meaning the distance from the center to this line is equal to the radius \( r \).

Therefore:
\[
|h - 12| = r
\]
But since \( h = r \), we can write:
\[
|r - 12| = r
\]

### Step 3: Solve for \( r \)
This equation can be split into two cases:

1. **Case 1**: \( r - 12 = r \)  
   This simplifies to \( 0 = 12 \), which is a contradiction. So, this case is not valid.

2. **Case 2**: \( r - 12 = -r \)  
   Simplifying this, we get:
   \[
   2r = 12 \quad \Rightarrow \quad r = 6
   \]

### Step 4: Determine the Center
Since \( r = 6 \) and \( h = k = r \), the center of the circle is at \( (6, 6) \).

### Step 5: Write the Equation of the Circle
The general form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Substituting \( h = 6 \), \( k = 6 \), and \( r = 6 \), we get:
\[
(x - 6)^2 + (y - 6)^2 = 36
\]

### Step 6: Expand to General Form
Expanding the equation:
\[
(x^2 - 12x + 36) + (y^2 - 12y + 36) = 36
\]
\[
x^2 + y^2 - 12x - 12y + 72 = 36
\]
\[
x^2 + y^2 - 12x - 12y + 36 = 0
\]

### Final Answer
The equation of the circle in general form is:
\[
\boxed{x^2 + y^2 - 12x - 12y + 36 = 0}
\]

Would you like any further details or have any questions? Here are some related questions you might find useful:

1. How to find the equation of a circle given three points on its circumference?
2. What is the general form of the equation of a circle?
3. How do you find the radius of a circle from its equation?
4. What is the method to determine the center of a circle using tangents?
5. How to convert the standard form of a circle's equation to the general form?
6. What are tangent lines and how are they related to circles?
7. How to determine if a given line is tangent to a given circle?
8. How do the coordinates of a center affect the tangency of a circle?

**Tip:** Always remember that the distance from the center of a circle to a tangent line is equal to the radius of the circle.

Discover how to find the equation of a circle tangent to the positive x-axis, negative y-axis, and the line x = 12. This problem involves circle geometry and the equation of a circle, covering tangent lines and circle centers.

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