To determine which equation describes the circle, we need to use the standard form of the equation of a circle.
The standard form equation of a circle centered at [tex]\((h, k)\)[/tex] with radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Given:
- The center of the circle is [tex]\((-5, 7)\)[/tex], so [tex]\(h = -5\)[/tex] and [tex]\(k = 7\)[/tex].
- The radius is [tex]\(11\)[/tex], so [tex]\(r = 11\)[/tex].
Let's substitute these values into the standard form equation.
1. Plug the center coordinates [tex]\((-5, 7)\)[/tex] into the formula:
[tex]\[
(x - (-5))^2 + (y - 7)^2 = r^2
\][/tex]
Simplify the equation where needed:
[tex]\[
(x + 5)^2 + (y - 7)^2 = r^2
\][/tex]
2. Replace the radius [tex]\(r\)[/tex] with [tex]\(11\)[/tex]:
[tex]\[
(x + 5)^2 + (y - 7)^2 = 11^2
\][/tex]
3. Calculate the square of the radius:
[tex]\[
11^2 = 121
\][/tex]
Now substituting back, we get:
[tex]\[
(x + 5)^2 + (y - 7)^2 = 121
\][/tex]
So, the correct equation that describes the circle centered at [tex]\((-5, 7)\)[/tex] with a radius of [tex]\(11\)[/tex] is:
[tex]\[
(x + 5)^2 + (y - 7)^2 = 121
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[
(x + 5)^2 + (y - 7)^2 = 121
\][/tex]
