Certainly! Let's find the center and radius of the given circle whose equation is:
[tex]\[
(x + 13)^2 + (y + 2)^2 = 81
\][/tex]
To identify the center and radius, we need to compare this equation to the standard form of a circle's equation:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
In the standard form:
- [tex]\((h, k)\)[/tex] is the center of the circle.
- [tex]\(r\)[/tex] is the radius of the circle.
Now, let's identify the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r\)[/tex] by rewriting and comparing this information:
1. Identify the center [tex]\((h, k)\)[/tex]:
- From [tex]\((x + 13)^2\)[/tex], we see that [tex]\(x + 13\)[/tex] can be written as [tex]\(x - (-13)\)[/tex]. Therefore, [tex]\(h = -13\)[/tex].
- From [tex]\((y + 2)^2\)[/tex], we see that [tex]\(y + 2\)[/tex] can be written as [tex]\(y - (-2)\)[/tex]. Therefore, [tex]\(k = -2\)[/tex].
Hence, the center of the circle is [tex]\((-13, -2)\)[/tex].
2. Determine the radius [tex]\(r\)[/tex]:
- The right-hand side of the equation [tex]\((x + 13)^2 + (y + 2)^2 = 81\)[/tex] corresponds to [tex]\(r^2\)[/tex] in the standard form.
- Thus, [tex]\(r^2 = 81\)[/tex].
- To find the radius [tex]\(r\)[/tex], we take the square root of both sides: [tex]\(r = \sqrt{81}\)[/tex], so [tex]\(r = 9.0\)[/tex] (since the radius is a non-negative value).
Therefore, the center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] of the circle can be written as:
[tex]\[
\text{Center} = (-13, -2)
\][/tex]
[tex]\[
\text{Radius} = 9.0
\][/tex]
