To determine the equation of a circle, we use the standard form of the equation for a circle:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
In this equation:
- [tex]\((h, k)\)[/tex] are the coordinates of the center of the circle.
- [tex]\(r\)[/tex] is the radius of the circle.
Given the center of the circle is [tex]\((0, -2)\)[/tex] and the radius is [tex]\(5\)[/tex]:
1. Substitute [tex]\(h = 0\)[/tex], [tex]\(k = -2\)[/tex], and [tex]\(r = 5\)[/tex] into the standard form equation.
[tex]\[
(x - 0)^2 + (y - (-2))^2 = 5^2
\][/tex]
2. Simplify the substitution:
[tex]\[
x^2 + (y + 2)^2 = 25
\][/tex]
Therefore, the equation of the circle with center [tex]\((0, -2)\)[/tex] and radius [tex]\(5\)[/tex] is:
[tex]\[
\boxed{x^2 + (y + 2)^2 = 25}
\][/tex]
