To determine the equation of a circle, we use the standard form of the circle equation:
[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.
In this problem, the center of the circle is given as [tex]\((-5, 5)\)[/tex] and the radius is given as [tex]\(3\)[/tex] units. Let's substitute these values into the standard form equation:
1. Substitute the center coordinates: The center [tex]\((h, k)\)[/tex] is [tex]\((-5, 5)\)[/tex], so we have:
- [tex]\(h = -5\)[/tex]
- [tex]\(k = 5\)[/tex]
2. Substitute the radius: The radius [tex]\(r\)[/tex] is [tex]\(3\)[/tex], so [tex]\(r^2 = 3^2 = 9\)[/tex].
Now, substitute [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r^2\)[/tex] into the standard form equation:
[tex]\[
(x - (-5))^2 + (y - 5)^2 = 3^2
\][/tex]
Simplify the terms:
[tex]\[
(x + 5)^2 + (y - 5)^2 = 9
\][/tex]
Therefore, the equation that represents a circle with a center at [tex]\((-5, 5)\)[/tex] and a radius of 3 units is:
[tex]\[
(x + 5)^2 + (y - 5)^2 = 9
\][/tex]
Based on the given choices, the correct answer is:
C. [tex]\((x + 5)^2 + (y - 5)^2 = 9\)[/tex]
