To write the equation of a circle in standard form, we need two pieces of information: the center of the circle [tex]\((h, k)\)[/tex] and the radius [tex]\(r\)[/tex]. The standard form of the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Given:
- The center of the circle is [tex]\((-4, -5)\)[/tex]. This means [tex]\(h = -4\)[/tex] and [tex]\(k = -5\)[/tex].
- The area of the circle is [tex]\(400\pi\)[/tex].
First, let's determine the radius [tex]\(r\)[/tex] from the given area. The formula for the area [tex]\(A\)[/tex] of a circle in terms of its radius [tex]\(r\)[/tex] is:
[tex]\[
A = \pi r^2
\][/tex]
Given the area [tex]\(400\pi\)[/tex]:
[tex]\[
400\pi = \pi r^2
\][/tex]
To solve for [tex]\(r^2\)[/tex], we divide both sides of the equation by [tex]\(\pi\)[/tex]:
[tex]\[
400 = r^2
\][/tex]
Taking the square root of both sides, we find:
[tex]\[
r = \sqrt{400} = 20
\][/tex]
Now that we have the radius, we can substitute [tex]\(h = -4\)[/tex], [tex]\(k = -5\)[/tex], and [tex]\(r = 20\)[/tex] into the standard form equation of a circle:
[tex]\[
(x - (-4))^2 + (y - (-5))^2 = 20^2
\][/tex]
Simplifying the equation:
[tex]\[
(x + 4)^2 + (y + 5)^2 = 400
\][/tex]
Thus, the equation of the circle in standard form is:
[tex]\[
(x + 4)^2 + (y + 5)^2 = 400
\][/tex]
