To determine 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 general 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 its radius.
Now, let's compare the given equation with the standard form step-by-step:
1. Identify the parts of the standard form in the given equation:
[tex]\[
(x + 9)^2 + (y - 6)^2 = 10^2
\][/tex]
2. Recall that in the standard form [tex]\((x - h)\)[/tex] and [tex]\((y - k)\)[/tex], [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are the coordinates of the center, but with signs opposite to those in the equation due to the form [tex]\((x - h)\)[/tex] and [tex]\((y - k)\)[/tex].
3. For [tex]\(x\)[/tex]:
- In [tex]\((x - h)^2\)[/tex], we have [tex]\((x + 9)^2\)[/tex] in our given equation.
- This indicates [tex]\(x - h = x + 9\)[/tex], which means [tex]\(h = -9\)[/tex].
4. For [tex]\(y\)[/tex]:
- In [tex]\((y - k)^2\)[/tex], we have [tex]\((y - 6)^2\)[/tex] in our given equation.
- This indicates [tex]\(y - k = y - 6\)[/tex], which means [tex]\(k = 6\)[/tex].
So, the center [tex]\((h, k)\)[/tex] of the circle is [tex]\((-9, 6)\)[/tex].
Thus, the correct answer is:
[tex]\[
(-9, 6)
\][/tex]
This result matches the provided options, confirming that the center of the circle is [tex]\((-9, 6)\)[/tex].
