To determine the equation of a circle with a given center and radius, we use the standard form of the circle's equation:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given:
- The center of the circle \((h, k) = (3.2, -2.1)\)
- The radius \(r = 4.3\)
We substitute these values into the standard form equation:
1. Replace \(h\) with \(3.2\):
[tex]\[
(x - 3.2)^2
\][/tex]
2. Replace \(k\) with \(-2.1\):
[tex]\[
(y - (-2.1))^2 = (y + 2.1)^2
\][/tex]
3. Replace \(r\) with \(4.3\):
[tex]\[
(4.3)^2 = 18.49
\][/tex]
Thus, the equation becomes:
[tex]\[
(x - 3.2)^2 + (y + 2.1)^2 = 18.49
\][/tex]
Next, we match this equation to the given options:
- Option A: [tex]$(x+3.2)^2+(y-2.1)^2=4.3$[/tex]
- This does not match our derived equation, as it does not correctly represent the center and the radius squared is not correct.
- Option B: [tex]$(x+2.1)^2+(y-3.2)^2=8.6$[/tex]
- The placement of the center coordinates is incorrect and the radius value is not squared.
- Option C: [tex]$(x-2.1)^2-(y+3.2)^2=(4.3)^2$[/tex]
- The circle’s center coordinates are reversed and it has an inappropriate subtraction operator.
- Option D: [tex]$(x-3.2)^2+(y+2.1)^2=(4.3)^2$[/tex]
- This correctly represents the center \((3.2, -2.1)\) and the radius squared.
Therefore, the correct equation of the circle is:
[tex]\[
\boxed{(x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2}
\][/tex]
