To determine which equation represents a circle with a center at [tex]\((-5, 5)\)[/tex] and a radius of 3 units, we'll use the standard form of the equation of a circle. Here's a step-by-step explanation:
1. Standard Form of Circle Equation:
The general equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
2. Identify Given Values:
- Center [tex]\((h, k) = (-5, 5)\)[/tex]
- Radius [tex]\(r = 3\)[/tex]
3. Substitute the Values:
Substitute [tex]\(h = -5\)[/tex], [tex]\(k = 5\)[/tex], and [tex]\(r = 3\)[/tex] into the standard equation:
[tex]\[
(x - (-5))^2 + (y - 5)^2 = 3^2
\][/tex]
4. Simplify the Equation:
Simplify the equation step-by-step:
[tex]\[
(x + 5)^2 + (y - 5)^2 = 9
\][/tex]
So, the correct equation representing the circle with center [tex]\((-5, 5)\)[/tex] and radius 3 is:
[tex]\[
(x + 5)^2 + (y - 5)^2 = 9
\][/tex]
Among the given choices, the equation that matches this form is:
A. [tex]\((x+5)^2 + (y-5)^2 = 9\)[/tex]
Therefore, the correct answer is [tex]\(\boxed{A}\)[/tex].
