Question

Identify the center and radius of the circle represented by the equation x squared plus y squared minus 2 x plus 6 y plus 6 equals 0.
(1 point)
Responses

center: left parenthesis 1 comma negative 3 right parenthesis radius: 2
center: Image with alt text: left parenthesis 1 comma negative 3 right parenthesis radius: Image with alt text: 2

center: left parenthesis 1 comma negative 3 right parenthesis radius: 4
center: Image with alt text: left parenthesis 1 comma negative 3 right parenthesis radius: Image with alt text: 4

center: left parenthesis negative 1 comma 3 right parenthesis radius: 2
center: Image with alt text: left parenthesis negative 1 comma 3 right parenthesis radius: Image with alt text: 2

center: left parenthesis negative 1 comma 3 right parenthesis radius: 4

Answer 1

To find the center and radius of the circle represented by the equation \(x^2 + y^2 - 2x + 6y + 6 = 0\), we can rewrite the equation in the standard form of a circle.

The standard form of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Let's start with the original equation:

\[
x^2 + y^2 - 2x + 6y + 6 = 0
\]

We can rearrange it to:

\[
x^2 - 2x + y^2 + 6y + 6 = 0
\]

Next, we will complete the square for both \(x\) and \(y\).

### Completing the square for \(x\):

1. Take the coefficient of \(x\) (which is \(-2\)), divide by 2 (giving \(-1\)), and square it (giving \(1\)):

\[
x^2 - 2x = (x - 1)^2 - 1
\]

### Completing the square for \(y\):

1. Take the coefficient of \(y\) (which is \(6\)), divide by 2 (giving \(3\)), and square it (giving \(9\)):

\[
y^2 + 6y = (y + 3)^2 - 9
\]

### Substitute back into the equation:

Now substituting these back into the equation gives us:

\[
(x - 1)^2 - 1 + (y + 3)^2 - 9 + 6 = 0
\]

Simplifying this:

\[
(x - 1)^2 + (y + 3)^2 - 4 = 0
\]
\[
(x - 1)^2 + (y + 3)^2 = 4
\]

Now we can identify the center and radius from this standard form:

- The center \((h, k)\) is \((1, -3)\).
- The radius \(r\) is \(\sqrt{4} = 2\).

Thus, the center of the circle is \((1, -3)\) and the radius is \(2\).

The correct response is:

**center: (1, -3) radius: 2**