To find the equation of a circle given its center and radius, we use the standard formula for 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
- [tex]\(r\)[/tex] is the radius of the circle
For this specific problem:
- The center of the circle is [tex]\((-3.2, 2.1)\)[/tex], thus [tex]\(h = -3.2\)[/tex] and [tex]\(k = 2.1\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(4.3\)[/tex].
Substituting these values into the standard formula, we get:
[tex]\[
(x - (-3.2))^2 + (y - 2.1)^2 = (4.3)^2
\][/tex]
Simplifying the equation, notice that subtracting a negative number is the same as adding the positive:
[tex]\[
(x + 3.2)^2 + (y - 2.1)^2 = (4.3)^2
\][/tex]
Now, let's compare this derived equation with the given options:
A. [tex]\((x + 2.1)^2 + (y - 3.2)^2 = 4.3^2\)[/tex]
B. [tex]\((x - 2.1)^2 - (y + 3.2)^2 = (4.3)^2\)[/tex]
C. [tex]\((x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2\)[/tex]
D. [tex]\((x + 3.2)^2 + (y - 2.1)^2 = 4.3^2\)[/tex]
Clearly, the correct equation matches option D:
[tex]\[
(x + 3.2)^2 + (y - 2.1)^2 = 4.3^2
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
\boxed{(x + 3.2)^2 + (y - 2.1)^2 = 4.3^2}
\][/tex]
The correct answer is option D.
