To determine the center of the circle given by the equation [tex]\((x + 5)^2 + (y - 8)^2 = 1\)[/tex], we need to compare this equation to the standard form of a circle's equation, which is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
In the standard form, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] is the radius of the circle.
Given the equation:
[tex]\[
(x + 5)^2 + (y - 8)^2 = 1
\][/tex]
we can rewrite the terms inside the parentheses to match the standard form [tex]\((x - h)\)[/tex] and [tex]\((y - k)\)[/tex].
Notice that:
[tex]\[
(x + 5)^2 = (x - (-5))^2
\][/tex]
and
[tex]\[
(y - 8)^2
\][/tex]
Thus, comparing this to the standard form [tex]\((x - h)^2 + (y - k)^2\)[/tex], we see that:
[tex]\[
h = -5, \quad k = 8
\][/tex]
So, the center of the circle is:
[tex]\((h, k) = (-5, 8)\)[/tex]
Therefore, the correct answer is:
D. [tex]\((-5, 8)\)[/tex]
