To determine the coordinates of the center of the circle given by 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 standard form of a circle's equation is given by:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
In this equation:
- [tex]\((h, k)\)[/tex] are the coordinates of the center of the circle.
- [tex]\(r\)[/tex] is the radius of the circle.
Let’s identify [tex]\(h\)[/tex] and [tex]\(k\)[/tex] in the given equation:
[tex]\[ (x - 6)^2 + (y + 5)^2 = 15^2 \][/tex]
Here, we compare the given equation with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
1. Notice that [tex]\(x\)[/tex] is modified by [tex]\(6\)[/tex] in the term [tex]\((x - 6)\)[/tex]. This means [tex]\(h = 6\)[/tex].
2. Notice that [tex]\(y\)[/tex] is modified by [tex]\(5\)[/tex] in the term [tex]\((y + 5)\)[/tex]. Since the standard form is [tex]\((y - k)\)[/tex], we need to rewrite [tex]\((y + 5)\)[/tex] in the same form:
[tex]\[ y + 5 = y - (-5) \][/tex]
This shows that [tex]\(k = -5\)[/tex].
Thus, the coordinates of the center of the circle are [tex]\((h, k) = (6, -5)\)[/tex].
Therefore, the correct choice is:
C. [tex]\((6, -5)\)[/tex]
