To find the coordinates of the center of the circle given the equation [tex]\((x - 6)^2 + (y + 5)^2 = 15^2\)[/tex], we need to recognize the standard form of a circle's equation. The general form of the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] is the radius.
For the given equation [tex]\((x - 6)^2 + (y + 5)^2 = 15^2\)[/tex], we can identify [tex]\(h\)[/tex] and [tex]\(k\)[/tex] by comparing it with the standard form. Let's break it down:
1. [tex]\((x - 6)^2\)[/tex] corresponds to the term [tex]\((x - h)^2\)[/tex]. Thus, [tex]\(h = 6\)[/tex].
2. [tex]\((y + 5)^2\)[/tex] can be rewritten as [tex]\((y - (-5))^2\)[/tex], which aligns with [tex]\((y - k)^2\)[/tex]. Therefore, [tex]\(k = -5\)[/tex].
From this, we determine that the center of the circle [tex]\((h, k)\)[/tex] is at the coordinates [tex]\((6, -5)\)[/tex].
Hence, the correct answer is:
D. [tex]\((6, -5)\)[/tex]
