To identify the center of the circle defined by the given equation:
[tex]\[ x^2 - 14x + y^2 + 14y = -82 \][/tex]
we need to rewrite the equation in the standard form of a circle:
[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. To do that, we need to complete the square for both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
### Completing the Square for the [tex]\(x\)[/tex]-terms
Start with the quadratic expression in [tex]\(x\)[/tex]:
[tex]\[ x^2 - 14x \][/tex]
To complete the square, add and subtract [tex]\((\frac{-14}{2})^2 = 49\)[/tex]:
[tex]\[ x^2 - 14x + 49 - 49 \][/tex]
This can be rewritten as:
[tex]\[ (x - 7)^2 - 49 \][/tex]
### Completing the Square for the [tex]\(y\)[/tex]-terms
Next, consider the quadratic expression in [tex]\(y\)[/tex]:
[tex]\[ y^2 + 14y \][/tex]
Similarly, add and subtract [tex]\((\frac{14}{2})^2 = 49\)[/tex]:
[tex]\[ y^2 + 14y + 49 - 49 \][/tex]
This can be rewritten as:
[tex]\[ (y + 7)^2 - 49 \][/tex]
### Rewriting the Original Equation
Now, rewrite the original equation incorporating these completed squares:
[tex]\[ (x - 7)^2 - 49 + (y + 7)^2 - 49 = -82 \][/tex]
Combine the constants on the right side:
[tex]\[
(x - 7)^2 + (y + 7)^2 - 98 = -82
\][/tex]
Add 98 to both sides to isolate the completed squares:
[tex]\[
(x - 7)^2 + (y + 7)^2 = 16
\][/tex]
### Identifying the Center and Radius
The equation is now in the standard form of a circle, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((7, -7)\)[/tex] and the radius [tex]\(r\)[/tex] is [tex]\(\sqrt{16} = 4\)[/tex].
Therefore, the center of the circle is:
[tex]\[
\boxed{(7, -7)}
\][/tex]
So the correct answer is D.
