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**Problem 12:** 

The question appears to be related to the equation of a circle:

"Persamaan lingkaran yang berpusat pada garis \( x + y = 4 \) dan menyinggung sumbu \( X \) adalah:
\( x^2 + y^2 - 4x - 6y + C = 0 \). Jumlah nilai \( C \) yang mungkin adalah...?"

This problem is asking for the possible values of \( C \) given that the circle touches the X-axis and has its center on the line \( x + y = 4 \).

### Step-by-Step Solution:

1. **Equation of the Circle:**
   The standard form of the equation of a circle is:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Expanding this, we get:
   \[
   x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = 0
   \]
   Comparing with the given equation:
   \[
   x^2 + y^2 - 4x - 6y + C = 0
   \]
   We can identify that:
   \[
   h = 2, \quad k = 3, \quad h^2 + k^2 - r^2 = C
   \]

2. **Condition for the Center on the Line:**
   The center \((h, k)\) lies on the line \(x + y = 4\). So:
   \[
   2 + 3 = 4 \quad \text{(This doesn't hold; there's a misunderstanding)}
   \]
   Actually, we need to find where the center lies based on \( x + y = 4 \) and then compute \( r^2 = k^2 \) since it touches the X-axis.

3. **For a circle that touches the X-axis:**
   The radius \( r = k \) (because distance from the center to the X-axis is \( k \), and this distance is the radius when it touches the X-axis).

4. **Compute the potential values for \( C \):**
   The general equation relating to \( C \) is:
   \[
   C = h^2 + k^2 - r^2
   \]
   Here \( r = k \), so:
   \[
   C = h^2 + k^2 - k^2 = h^2
   \]

   Now, using the fact that \( h \) and \( k \) satisfy \( x + y = 4 \):
   Let \( h = x \), \( k = 4 - x \).

   \[
   C = h^2 = x^2
   \]

   Plugging different values for \( h \), we could compute the possible values for \( C \).

Therefore, to solve it completely and get the precise values for \( C \), you would plug in possible \( h \) values into the equation above.

If more details are required, I can break down the steps with specific values for \( h \). Would you like to continue with this approach, or is there anything specific you'd like clarified?

Explore how to solve the equation of a circle problem involving the center on the line x + y = 4 and touching the X-axis. This problem requires understanding circle equations and coordinate geometry concepts. Suitable for Grades 10-12.

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