To determine the center of a circle represented by the given equation [tex]\((x + 9)^2 + (y - 6)^2 = 10^2\)[/tex], we need to follow these steps:
1. Recall the general form of the circle equation:
The general form of the equation of a circle is [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 of the circle.
2. Compare the given equation with the general form:
The given equation is [tex]\((x + 9)^2 + (y - 6)^2 = 10^2\)[/tex]. To match it to the general form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we need to rewrite it in a comparable structure.
3. Identify the center coordinates:
- For [tex]\((x + 9)^2\)[/tex], we see it matches [tex]\((x - (-9))^2\)[/tex]. This means [tex]\(h = -9\)[/tex].
- For [tex]\((y - 6)^2\)[/tex], it is already in the form [tex]\((y - k)^2\)[/tex], so [tex]\(k = 6\)[/tex].
4. Determine the center:
Thus, the center of the circle is [tex]\((-9, 6)\)[/tex].
Therefore, the center of the circle represented by the equation [tex]\((x + 9)^2 + (y - 6)^2 = 10^2\)[/tex] is [tex]\((-9, 6)\)[/tex].
The correct answer is [tex]\(\boxed{(-9, 6)}\)[/tex].
