To find the center and the radius of the circle given by the equation [tex]\(x^2 + y^2 + 6x - 2y + 1 = 0\)[/tex], we can follow a step-by-step process involving the method known as "completing the square."
1. Rearrange the equation: First, we group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms together:
[tex]\[
x^2 + 6x + y^2 - 2y + 1 = 0
\][/tex]
2. Complete the square for [tex]\(x\)[/tex]:
- Look at the [tex]\(x\)[/tex]-terms: [tex]\(x^2 + 6x\)[/tex]
- To complete the square, we add and subtract [tex]\(\left(\frac{6}{2}\right)^2 = 9\)[/tex]:
[tex]\[
x^2 + 6x + 9 - 9
\][/tex]
- This can be rewritten as:
[tex]\[
(x + 3)^2 - 9
\][/tex]
3. Complete the square for [tex]\(y\)[/tex]:
- Look at the [tex]\(y\)[/tex]-terms: [tex]\(y^2 - 2y\)[/tex]
- To complete the square, we add and subtract [tex]\(\left(\frac{-2}{2}\right)^2 = 1\)[/tex]:
[tex]\[
y^2 - 2y + 1 - 1
\][/tex]
- This can be rewritten as:
[tex]\[
(y - 1)^2 - 1
\][/tex]
4. Rewrite the original equation with these completed squares:
[tex]\[
(x + 3)^2 - 9 + (y - 1)^2 - 1 + 1 = 0
\][/tex]
Combining the constants, we get:
[tex]\[
(x + 3)^2 + (y - 1)^2 - 9 = 0
\][/tex]
Simplifying further, we obtain:
[tex]\[
(x + 3)^2 + (y - 1)^2 = 9
\][/tex]
5. Identify the center and radius:
- The equation [tex]\((x + 3)^2 + (y - 1)^2 = 9\)[/tex] is in standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
- Comparing, we see that [tex]\(h = -3\)[/tex] and [tex]\(k = 1\)[/tex], so the center of the circle is [tex]\((-3, 1)\)[/tex].
- The radius can be determined from [tex]\(r^2 = 9\)[/tex], so [tex]\(r = \sqrt{9} = 3\)[/tex].
Conclusion: The coordinates of the center of circle [tex]\(M\)[/tex] are [tex]\((-3, 1)\)[/tex], and the length of the radius is [tex]\(3\)[/tex].
