$$ 1 \leftarrow \left\lvert\, \begin{array}{l} ext { A circle has the equation } x^{2}+y^{2}-3 x-6 y+9=0 . \ ext { (a) Find the center (h,k) and radius } r ext { of the circle. } \ ext { (b) Graph the circle. } \ ext { (c) Find the intercepts, if ahy, of the graph. } \ \begin{array}{l} ext { (a) The center of the circle is } \ ext { (Type an ordered pair, using integers or fractions.) }\end{array}\end{array} ight. $$

Question

$$ 1 \leftarrow \left\lvert\, \begin{array}{l}	ext { A circle has the equation } x^{2}+y^{2}-3 x-6 y+9=0 . \ 	ext { (a) Find the center (h,k) and radius } r 	ext { of the circle. } \ 	ext { (b) Graph the circle. } \ 	ext { (c) Find the intercepts, if ahy, of the graph. } \ \begin{array}{l}	ext { (a) The center of the circle is } \ 	ext { (Type an ordered pair, using integers or fractions.) }\end{array}\end{array}ight. $$

UpStudy AI · Accepted Answer

To find the center $$(h, k)$$ and the radius $$r$$ of the circle given by the equation:

$$
x^{2} + y^{2} - 3x - 6y + 9 = 0
$$

we can follow these steps:

### **Step 1: Rewrite the Equation in Standard Form**

The standard form of a circle's equation is:

$$
(x - h)^2 + (y - k)^2 = r^2
$$

To convert the given equation to this form, we'll complete the square for both $$x$$ and $$y$$.

### **Step 2: Complete the Square for $$x$$ and $$y$$**

1. **Group the $$x$$ and $$y$$ terms:**

$$
   x^{2} - 3x + y^{2} - 6y = -9
   $$

2. **Complete the square for $$x$$:**

$$
   x^{2} - 3x = \left(x - \frac{3}{2}\right)^2 - \frac{9}{4}
   $$

3. **Complete the square for $$y$$:**

$$
   y^{2} - 6y = (y - 3)^2 - 9
   $$

4. **Substitute these back into the equation:**

$$
   \left(x - \frac{3}{2}\right)^2 - \frac{9}{4} + (y - 3)^2 - 9 = -9
   $$

5. **Combine the constants:**

$$
   \left(x - \frac{3}{2}\right)^2 + (y - 3)^2 = \frac{9}{4}
   $$

### **Step 3: Identify the Center and Radius**

From the standard form:

$$
(x - h)^2 + (y - k)^2 = r^2
$$

We can see that:

- **Center $$(h, k)$$:** $$\left(\frac{3}{2},\ 3\right)$$
- **Radius $$r$$:** $$\frac{3}{2}$$

### **Final Answer**

**(a) The center of the circle is $$\left( \frac{3}{2},\ 3 \right)$$ and the radius is $$\frac{3}{2}$$.**
The center of the circle is $$\left( \frac{3}{2},\ 3 \right)$$ and the radius is $$\frac{3}{2}$$.

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