To determine the equation of the circle with a given center and radius, we use the standard form of the equation of a circle.
The standard form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is given by:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Given:
- Center [tex]\((h, k) = (3.2, -2.1)\)[/tex]
- Radius [tex]\(r = 4.3\)[/tex]
Substituting the given values into the standard form, we obtain:
[tex]\[
(x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2
\][/tex]
Now let's examine the given options to see which one matches this equation:
- Option A: [tex]\((x+2.1)^2+(y-3.2)^2=8.6\)[/tex]
This does not match our equation, as the signs of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are reversed, and the right-hand side [tex]\(8.6\)[/tex] does not equal [tex]\((4.3)^2\)[/tex].
- Option B: [tex]\((x+3.2)^2+(y-2.1)^2=4.3\)[/tex]
This does not match our equation, as the signs of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are reversed, and the right-hand side [tex]\(4.3\)[/tex] does not equal [tex]\((4.3)^2\)[/tex].
- Option C: [tex]\((x-3.2)^2+(y+2.1)^2=(4.3)^2\)[/tex]
This matches our equation exactly: [tex]\((x - 3.2)^2 + (y + 2.1)^2 = (4.3)^2\)[/tex].
- Option D: [tex]\((x-2.1)^2-(y+3.2)^2=(4.3)^2\)[/tex]
This does not match our equation, as the values of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are incorrect, and there is a subtraction symbol in the equation, which is not part of the standard form for a circle.
Therefore, the correct equation of the circle is given in option C:
[tex]\[
(x-3.2)^2+(y+2.1)^2=(4.3)^2
\][/tex]
So the correct answer is:
Option C
