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).

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.