Let's rewrite the given equation:
[tex]\[ x^2 + 16x + y^2 - 16y = -112 \][/tex]
We aim to rewrite it in the center-radius form of the equation of a circle, which has the form:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Here are the steps to achieve this form:
1. Group and complete the square for the [tex]\(x\)[/tex] terms:
[tex]\[
x^2 + 16x
\][/tex]
Completing the square involves adding and subtracting the same value. To complete the square for [tex]\(x^2 + 16x\)[/tex]:
- Take half of 16, which is 8.
- Square it to get 64.
So,
[tex]\[
x^2 + 16x = (x + 8)^2 - 64
\][/tex]
2. Group and complete the square for the [tex]\(y\)[/tex] terms:
[tex]\[
y^2 - 16y
\][/tex]
Similarly, complete the square for [tex]\(y^2 - 16y\)[/tex]:
- Take half of -16, which is -8.
- Square it to get 64.
So,
[tex]\[
y^2 - 16y = (y - 8)^2 - 64
\][/tex]
3. Substitute back into the original equation:
The equation now looks like this:
[tex]\[
(x + 8)^2 - 64 + (y - 8)^2 - 64 = -112
\][/tex]
4. Combine like terms:
[tex]\[
(x + 8)^2 + (y - 8)^2 - 128 = -112
\][/tex]
5. Solve for the constant term:
Adding 128 to both sides gives:
[tex]\[
(x + 8)^2 + (y - 8)^2 = 16
\][/tex]
Thus, we have successfully rewritten the given equation in the center-radius form of a circle:
[tex]\[
(x + 8)^2 + (y - 8)^2 = 16
\][/tex]
Comparing this with the given multiple choice options, we find that the correct answer is:
[tex]\[ \text{B. } (x + 8)^2 + (y - 8)^2 = 16 \][/tex]
