To find the equation of a circle, you can use the standard form:
[tex]\[(x - h)^2 + (y - k)^2 = r^2\][/tex]
where [tex]\((h, k)\)[/tex] are the coordinates of the center of the circle and [tex]\(r\)[/tex] is the radius.
Given the center is [tex]\((-4, 11)\)[/tex] and the radius is [tex]\(\sqrt{5}\)[/tex], the standard form of the equation of the circle is:
[tex]\[(x + 4)^2 + (y - 11)^2 = (\sqrt{5})^2\][/tex]
This can be simplified to:
[tex]\[(x + 4)^2 + (y - 11)^2 = 5\][/tex]
Now let's expand this equation:
[tex]\[
(x + 4)^2 + (y - 11)^2 = 5
\][/tex]
[tex]\[
= (x^2 + 8x + 16) + (y^2 - 22y + 121) = 5
\][/tex]
Combine the terms:
[tex]\[
x^2 + 8x + 16 + y^2 - 22y + 121 = 5
\][/tex]
Simplify by moving all terms to one side of the equation:
[tex]\[
x^2 + 8x + y^2 - 22y + 137 = 5
\][/tex]
[tex]\[
x^2 + 8x + y^2 - 22y + 132 = 0
\][/tex]
So, the final equation of the circle in its general form is:
[tex]\[
x^2 + y^2 + 8x - 22y + 132 = 0
\][/tex]
Therefore, the correct equation is:
[tex]\[
x^2 + y^2 + 8x - 22y + 132 = 0
\][/tex]
