Let's analyze the given equation of the circle step by step to find its center.
The standard 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 of the circle and [tex]\(r\)[/tex] is the radius of the circle.
We are given the equation:
[tex]\[
(x + 9)^2 + (y - 6)^2 = 10^2
\][/tex]
Our goal is to rewrite this given equation in the standard form and identify the values of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] which represent the coordinates of the center of the circle.
Notice that in the given equation, we have:
[tex]\[
(x + 9)^2
\][/tex]
This can be rewritten as:
[tex]\[
(x - (-9))^2
\][/tex]
indicating that [tex]\(h = -9\)[/tex].
Similarly, we have:
[tex]\[
(y - 6)^2
\][/tex]
which is already in the form [tex]\((y - k)^2\)[/tex] with [tex]\(k = 6\)[/tex].
Thus, by comparing the given equation [tex]\((x + 9)^2 + (y - 6)^2 = 10^2\)[/tex] with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can see:
- The center [tex]\(h\)[/tex] is [tex]\(-9\)[/tex]
- The center [tex]\(k\)[/tex] is [tex]\(6\)[/tex]
Therefore, the coordinates of the center of the circle are [tex]\((-9, 6)\)[/tex].
Among the given options, the correct answer is:
[tex]\[
(-9, 6)
\][/tex]
