To find the equation of a circle given its center and 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.
Let's analyze the given problem:
- Center of the circle [tex]\((h, k) = (-5, 5)\)[/tex]
- Radius [tex]\(r = 3\)[/tex]
Substituting the center and radius into the standard form equation, we get:
[tex]\[
(x - (-5))^2 + (y - 5)^2 = 3^2
\][/tex]
Simplify this equation:
[tex]\[
(x + 5)^2 + (y - 5)^2 = 9
\][/tex]
Thus, the equation of the circle is:
[tex]\[
(x + 5)^2 + (y - 5)^2 = 9
\][/tex]
Now, we compare this result with the provided options:
- A. [tex]\((x-5)^2+(y+5)^2=9\)[/tex] does not match our derived equation because the signs inside the parentheses are incorrect.
- B. [tex]\((x+5)^2+(y-5)^2=6\)[/tex] does not match our derived equation because the right-hand side is 6 instead of 9.
- C. [tex]\((x+5)^2+(y-5)^2=9\)[/tex] exactly matches our derived equation.
- D. [tex]\((x-5)^2+(y+5)^2=3\)[/tex] does not match our derived equation because both the left-hand side and right-hand side are incorrect.
- E. [tex]\((x+5)^2+(y-5)^2=3\)[/tex] does not match our derived equation because the right-hand side is 3 instead of 9.
Therefore, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
