To determine the equation of a circle given the center and the radius, we use the standard form of the equation of a circle:
[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.
Given:
- The center of the circle is [tex]\((-3, 2)\)[/tex]
- The radius of the circle is [tex]\(5\)[/tex]
First, let's substitute the given center [tex]\((-3, 2)\)[/tex] into the general equation. Here, [tex]\(h = -3\)[/tex] and [tex]\(k = 2\)[/tex]:
[tex]\[
(x - (-3))^2 + (y - 2)^2 = r^2
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
(x + 3)^2 + (y - 2)^2 = r^2
\][/tex]
Next, we substitute the given radius [tex]\(r = 5\)[/tex] into the equation:
[tex]\[
(x + 3)^2 + (y - 2)^2 = 5^2
\][/tex]
Since [tex]\(5^2 = 25\)[/tex], the equation becomes:
[tex]\[
(x + 3)^2 + (y - 2)^2 = 25
\][/tex]
Thus, the correct equation of the circle is:
[tex]\[
(x + 3)^2 + (y - 2)^2 = 25
\][/tex]
Therefore, the correct answer is [tex]\(D\)[/tex]:
[tex]\[
(x + 3)^2 + (y - 2)^2 = 25
\][/tex]
