To determine the correct equation for a circle with its center at [tex]\((2, -8)\)[/tex] and a radius of 11, let's use the standard form of the equation for a circle. The formula for the equation of a circle in standard form is:
[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.
Given:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((2, -8)\)[/tex],
- The radius [tex]\(r\)[/tex] is 11.
First, substitute [tex]\(h = 2\)[/tex], [tex]\(k = -8\)[/tex], and [tex]\(r = 11\)[/tex] into the standard form equation:
[tex]\[
(x - 2)^2 + (y + 8)^2 = 11^2
\][/tex]
Next, calculate [tex]\(11^2\)[/tex]:
[tex]\[
11^2 = 121
\][/tex]
So, the equation of the circle becomes:
[tex]\[
(x - 2)^2 + (y + 8)^2 = 121
\][/tex]
Let's now check which of the given options matches this equation:
1. [tex]\((x - 8)^2 + (y + 2)^2 = 11\)[/tex]
2. [tex]\((x - 2)^2 + (y + 8)^2 = 121\)[/tex]
3. [tex]\((x + 2)^2 + (y - 8)^2 = 11\)[/tex]
4. [tex]\((x + 8)^2 + (y - 2)^2 = 121\)[/tex]
Upon review, we see that the second option matches our derived equation:
[tex]\((x - 2)^2 + (y + 8)^2 = 121\)[/tex]
Therefore, the correct equation representing a circle with a center at [tex]\((2, -8)\)[/tex] and a radius of 11 is:
[tex]\[
(x - 2)^2 + (y + 8)^2 = 121
\][/tex]
