To find the center of the circle represented by the equation [tex]\((x + 9)^2 + (y - 6)^2 = 10^2\)[/tex], we need to understand the standard 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]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
Given the equation:
[tex]\[
(x + 9)^2 + (y - 6)^2 = 10^2
\][/tex]
We can rewrite the equation in the standard form by recognizing the shifts in the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates.
1. The term [tex]\((x + 9)^2\)[/tex] can be rewritten as [tex]\((x - (-9))^2\)[/tex]. This indicates that the [tex]\(x\)[/tex]-coordinate of the center is [tex]\(-9\)[/tex].
2. The term [tex]\((y - 6)^2\)[/tex] matches the standard form [tex]\((y - k)\)[/tex], where [tex]\(k = 6\)[/tex].
Therefore, the center [tex]\((h, k)\)[/tex] of the circle is [tex]\((-9, 6)\)[/tex].
Thus, the correct answer is:
[tex]\[
(-9, 6)
\][/tex]
