https://file.aiquickdraw.com/mathbot/file/17284567126997avaksai.jpg

The problem involves a circle equation and asks for the correct statements based on the given circle equation \( x^2 + y^2 - 6x + 2y + 9 = 0 \). Let's break down the steps:

### Step 1: Rewrite the equation
The equation is:
\[
x^2 + y^2 - 6x + 2y + 9 = 0
\]
First, we complete the square for both \( x \) and \( y \) terms.

- For the \( x \)-terms: \( x^2 - 6x \). 
  \[
  x^2 - 6x = (x - 3)^2 - 9
  \]

- For the \( y \)-terms: \( y^2 + 2y \).
  \[
  y^2 + 2y = (y + 1)^2 - 1
  \]

Now, substituting these into the original equation:
\[
(x - 3)^2 - 9 + (y + 1)^2 - 1 + 9 = 0
\]
Simplify:
\[
(x - 3)^2 + (y + 1)^2 - 1 = 0
\]
\[
(x - 3)^2 + (y + 1)^2 = 1
\]

### Step 2: Interpret the equation
This is the standard form of a circle:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \( (h, k) \) is the center and \( r \) is the radius. From the equation, we see that:

- The center is \( (3, -1) \).
- The radius is \( r = 1 \).

### Step 3: Analyze the statements
Now, we evaluate the given statements in the image:

1. **Koordinat pusat lingkaran adalah (3, -1)**: This is **true**. We found the center to be \( (3, -1) \).
2. **Jari-jari lingkaran adalah 10**: This is **false**. The radius is 1, not 10.
3. **Lingkaran menyinggung sumbu x**: This is **true**. Since the radius is 1, and the center is at \( (3, -1) \), the circle touches the x-axis at \( y = 0 \).
4. **Lingkaran menyinggung sumbu y**: This is **false**. The center is 3 units away from the y-axis, and the radius is only 1, so it does not touch the y-axis.

### Step 4: Correct Answer
Based on the evaluations:
- Statements (1) and (3) are correct.
Thus, the correct option is **b. (1) dan (3)**.

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Would you like more details on how to complete the square or clarify the concept of a circle equation? Here are some related questions for further practice:

1. How do you complete the square for a general quadratic equation?
2. What is the geometric interpretation of a circle's center and radius?
3. How do you determine if a circle touches the x-axis or y-axis?
4. Can a circle have a negative radius? Why or why not?
5. How would the equation change if the circle were to move?

**Tip:** Always check the distance of the circle’s center to the axes when determining if it touches or intersects them.

Given the circle equation x^2 + y^2 - 6x + 2y + 9 = 0, determine which statements about the center, radius, and tangency with the x- and y-axes are correct.

This guide provides a step-by-step solution to a circle equation problem, covering completing the square, finding the center and radius, and determining tangency with axes. Ideal for high school students learning circle geometry and algebra.

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