To determine the equation of a circle centered at [tex]\((-3, 2)\)[/tex] with a radius of 2, we use the general 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.
For the given circle:
- The center [tex]\((h, k)\)[/tex] is [tex]\((-3, 2)\)[/tex].
- The radius [tex]\(r\)[/tex] is 2.
Substitute these values into the equation:
1. Replace [tex]\(h\)[/tex] with [tex]\(-3\)[/tex], [tex]\(k\)[/tex] with 2, and [tex]\(r\)[/tex] with 2.
[tex]\[
(x - (-3))^2 + (y - 2)^2 = 2^2
\][/tex]
2. Simplify the expressions inside the parentheses:
[tex]\[
(x + 3)^2 + (y - 2)^2 = 4
\][/tex]
Hence, the correct equation that represents this circle is:
[tex]\[
(x + 3)^2 + (y - 2)^2 = 4
\][/tex]
Among the given options, this matches:
[tex]\[
(x + 3)^2 + (y - 2)^2 = 4
\][/tex]
Thus, the correct equation for the circle is the third option:
[tex]\[
\boxed{(x + 3)^2 + (y - 2)^2 = 4}
\][/tex]
