To find the center and radius of the circle given by the equation [tex]\( x^2 - 7x + y^2 + y + 1 = 0 \)[/tex], we need to rewrite this equation in the standard form of a circle's equation: [tex]\( (x - h)^2 + (y - k)^2 = r^2 \)[/tex]. This can be done by completing the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
### Step-by-Step Solution
1. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms separately:
[tex]\[
x^2 - 7x + y^2 + y + 1 = 0
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex]-terms:
- Start with [tex]\( x^2 - 7x \)[/tex].
- Half of the coefficient of [tex]\(x\)[/tex] is [tex]\(-7 / 2 = -3.5\)[/tex].
- Square this value: [tex]\((-3.5)^2 = 12.25\)[/tex].
- Rewrite [tex]\( x^2 - 7x \)[/tex] as [tex]\( (x - 3.5)^2 - 12.25 \)[/tex].
3. Complete the square for the [tex]\(y\)[/tex]-terms:
- Start with [tex]\( y^2 + y \)[/tex].
- Half of the coefficient of [tex]\(y\)[/tex] is [tex]\(1 / 2 = 0.5\)[/tex].
- Square this value: [tex]\((0.5)^2 = 0.25\)[/tex].
- Rewrite [tex]\( y^2 + y \)[/tex] as [tex]\( (y + 0.5)^2 - 0.25 \)[/tex].
4. Substitute these completed squares back into the original equation:
- Replace [tex]\( x^2 - 7x \)[/tex] with [tex]\( (x - 3.5)^2 - 12.25 \)[/tex].
- Replace [tex]\( y^2 + y \)[/tex] with [tex]\( (y + 0.5)^2 - 0.25 \)[/tex].
[tex]\[
(x - 3.5)^2 - 12.25 + (y + 0.5)^2 - 0.25 + 1 = 0
\][/tex]
5. Simplify the equation:
- Combine the constants: [tex]\(-12.25 - 0.25 + 1 = -11.5\)[/tex].
[tex]\[
(x - 3.5)^2 + (y + 0.5)^2 - 11.5 = 0
\][/tex]
- Add [tex]\(11.5\)[/tex] to both sides to get the standard form:
[tex]\[
(x - 3.5)^2 + (y + 0.5)^2 = 11.5
\][/tex]
In this form, [tex]\((x - 3.5)^2 + (y + 0.5)^2 = 11.5\)[/tex], we can easily identify the center and radius:
- The center [tex]\((h, k)\)[/tex] is [tex]\((3.5, -0.5)\)[/tex].
- The radius [tex]\( r \)[/tex] is [tex]\(\sqrt{11.5}\)[/tex].
Thus, the center of the circle is:
[tex]\[
\boxed{(3.5, -0.5)}
\][/tex]
And the radius of the circle is:
[tex]\[
\boxed{3.391164991562634}
\][/tex]
