To determine the center of the circle given by the equation [tex]\( x^2 + y^2 + 4x - 8y + 11 = 0 \)[/tex], we need to rewrite the equation in the standard form of a circle's equation, [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.
1. Rewrite the given equation and group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[
x^2 + y^2 + 4x - 8y + 11 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex]-terms:
[tex]\[
x^2 + 4x \][/tex]
To complete the square, add and subtract [tex]\(4\)[/tex] inside the equation:
[tex]\[
x^2 + 4x + 4 - 4 = 0 \][/tex]
Factorize the trinomial:
[tex]\[
(x + 2)^2 - 4 \][/tex]
3. Complete the square for the [tex]\(y\)[/tex]-terms:
[tex]\[
y^2 - 8y \][/tex]
To complete the square, add and subtract [tex]\(16\)[/tex] inside the equation:
[tex]\[
y^2 - 8y + 16 - 16 = 0 \][/tex]
Factorize the trinomial:
[tex]\[
(y - 4)^2 - 16 \][/tex]
4. Substitute the completed squares back into the original equation:
[tex]\[
(x + 2)^2 - 4 + (y - 4)^2 - 16 + 11 = 0 \][/tex]
5. Simplify the equation:
[tex]\[
(x + 2)^2 + (y - 4)^2 - 9 = 0 \][/tex]
[tex]\[
(x + 2)^2 + (y - 4)^2 = 9 \][/tex]
Now, the equation is in the standard form, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\( h = -2 \)[/tex] and [tex]\( k = 4 \)[/tex].
Therefore, the center of the circle is [tex]\((-2, 4)\)[/tex].
The correct answer is:
[tex]\((-2, 4)\)[/tex]
