Complete the square and write the given equation in standard form. Then give the center and radius of the circle and graph the equation. $$ x^{2}+y^{2}-2 x-6 y-15=0 $$

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To rewrite the given equation in standard form and identify the center and radius of the circle, follow these steps:

### Given Equation:
$$ x^{2} + y^{2} - 2x - 6y - 15 = 0 $$

### Step 1: Group $$ x $$ and $$ y $$ Terms
$$
(x^{2} - 2x) + (y^{2} - 6y) = 15
$$

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

**For the $$ x $$-terms:**
1. Take the coefficient of $$ x $$, which is $$-2$$, divide by 2 to get $$-1$$, and square it to get $$1$$.
2. Add and subtract this value inside the equation.

$$
x^{2} - 2x + 1 - 1
$$

**For the $$ y $$-terms:**
1. Take the coefficient of $$ y $$, which is $$-6$$, divide by 2 to get $$-3$$, and square it to get $$9$$.
2. Add and subtract this value inside the equation.

$$
y^{2} - 6y + 9 - 9
$$

### Step 3: Rewrite the Equation with Completed Squares
$$
(x^{2} - 2x + 1) + (y^{2} - 6y + 9) - 1 - 9 = 15
$$
$$
(x - 1)^2 + (y - 3)^2 - 10 = 15
$$

### Step 4: Isolate the Completed Squares
$$
(x - 1)^2 + (y - 3)^2 = 25
$$

### Standard Form of the Circle:
$$
(x - h)^2 + (y - k)^2 = r^2
$$
Where:
- $$ (h, k) $$ is the center.
- $$ r $$ is the radius.

### Identifying the Center and Radius:
- **Center ($$ h, k $$)**: $$ (1, 3) $$
- **Radius ($$ r $$)**: $$ \sqrt{25} = 5 $$

### Final Standard Form:
$$
(x - 1)^2 + (y - 3)^2 = 5^2
$$

### Graph of the Equation:
To graph the circle:

1. **Plot the Center** at $$ (1, 3) $$.
2. **Draw a Circle** with a radius of $$5$$ units around the center.

**Visualization Steps:**
- **Horizontal Coordinate (x-axis):** Start at $$ x = 1 $$.
- **Vertical Coordinate (y-axis):** Start at $$ y = 3 $$.
- From the center $$ (1, 3) $$, move $$5$$ units in all directions to outline the circle.

Here's a rough sketch of the graph:

```
y
|
8|          *
7|        *   *
6|      *       *
5|     *         *
4|    *           *
3|----*-----C-----*---- x
2|    *           *
1|     *         *
0|      *       *
-1|        *   *
-2|          *
```

- **C** represents the center at $$ (1, 3) $$.
- The asterisks ($$*$$) outline the circle with radius $$5$$.

### Summary:
- **Standard Form:** $$ (x - 1)^2 + (y - 3)^2 = 25 $$
- **Center:** $$ (1, 3) $$
- **Radius:** $$ 5 $$
The equation $$ x^{2} + y^{2} - 2x - 6y - 15 = 0 $$ can be rewritten in standard form as $$ (x - 1)^2 + (y - 3)^2 = 25 $$. This represents a circle with center at $$ (1, 3) $$ and a radius of $$ 5 $$.

Complete the square and write the given equation in standard form. Then give the center and radius of the circle and graph the equation. \( x^{2}+y^{2}-2 x-6 y-15=0 \)

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