To determine the center of a circle from its equation, we need to understand the general form of a circle's equation. The standard form of a circle's equation is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
In this equation:
- [tex]\((h, k)\)[/tex] represents the center of the circle.
- [tex]\(r\)[/tex] represents the radius.
Given the equation:
[tex]\[
(x + 9)^2 + (y - 6)^2 = 10^2
\][/tex]
Our goal is to identify the values of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] that represent the center of the circle. To convert the given equation into the standard form, we recognize the following:
1. [tex]\((x + 9)^2\)[/tex] can be rewritten as [tex]\((x - (-9))^2\)[/tex]. This tells us that [tex]\(h = -9\)[/tex].
2. [tex]\((y - 6)^2\)[/tex] already matches the standard form, so [tex]\(k = 6\)[/tex].
Therefore, the center of the circle [tex]\((h, k)\)[/tex] is:
[tex]\[
(-9, 6)
\][/tex]
This matches one of the provided answer choices. Thus, the correct answer is:
[tex]\[
(-9, 6)
\][/tex]
