https://file.aiquickdraw.com/mathbot/file/1723561846594qzsljmol.jpg

Let's analyze the problem in the image.

### Problem Analysis
The problem is asking for the equation of a circle that is tangent to the x-axis and y-axis at a certain point. The coordinates of the center of the circle are provided as \( (h, k) \), and the equation of the circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]

Given in the problem:
- The circle is tangent to the x-axis and y-axis.
- The equation form provided is \( Ax + By + C = 0 \).

### Given Information:
- The center of the circle is at \( \left(\frac{3}{2}, \frac{3}{2}\right) \).
- The radius \( r \) of the circle equals the distance from the center to either axis, which is \( r = \frac{3}{2} \).

### Step-by-Step Solution:

1. **Equation of the Circle:**
   The general equation of the circle centered at \( \left(h, k\right) \) is:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Plugging in \( h = \frac{3}{2} \), \( k = \frac{3}{2} \), and \( r = \frac{3}{2} \):
   \[
   \left(x - \frac{3}{2}\right)^2 + \left(y - \frac{3}{2}\right)^2 = \left(\frac{3}{2}\right)^2
   \]
   Simplifying:
   \[
   \left(x - \frac{3}{2}\right)^2 + \left(y - \frac{3}{2}\right)^2 = \frac{9}{4}
   \]

2. **Expanding the Equation:**
   Expand the left-hand side:
   \[
   \left(x^2 - 3x + \frac{9}{4}\right) + \left(y^2 - 3y + \frac{9}{4}\right) = \frac{9}{4}
   \]
   Simplifying, we get:
   \[
   x^2 + y^2 - 3x - 3y + \frac{9}{4} + \frac{9}{4} - \frac{9}{4} = 0
   \]
   \[
   x^2 + y^2 - 3x - 3y + \frac{9}{4} = 0
   \]

   Multiply through by 4 to eliminate the fraction:
   \[
   4x^2 + 4y^2 - 12x - 12y + 9 = 0
   \]

3. **Converting to General Form:**
   Now, we compare it to the general form \( Ax + By + C = 0 \) and identify:
   \[
   A = 4, B = -12, C = 9
   \]

4. **Sum of All Possible Values of C:**
   Since the problem asks for the sum of all possible values of C, considering multiple possibilities, here it simplifies to a single possible value of \( C \), which is \( C = 9 \).

So, the **sum of all possible values of \( C \)** is **9**. 

Would you like to discuss more details, or do you have any questions?

### Related Questions:
1. How do you find the distance between a point and a line?
2. What is the equation of a circle tangent to both x-axis and y-axis?
3. How do you derive the general form of a circle equation?
4. What is the significance of the center and radius in a circle's equation?
5. How do you convert a quadratic equation to the standard form?
6. What is the relationship between the coefficients in the equation of a circle?
7. How can you determine the points of tangency for a circle and an axis?
8. How would the problem change if the circle were tangent to the x-axis at a different point?

### Tip:
When solving circle-related problems, always sketch a diagram if possible to visualize the relationship between the circle, its center, and the axes.

Learn how to find the equation of a circle tangent to both the x-axis and y-axis with a detailed step-by-step solution. This problem covers key concepts in geometry and coordinate geometry, including the equation of a circle and its properties. Suitable for students in Grades 10-12.

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