To determine the equation of a circle, we use the standard form of the circle's 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.
Given the center of the circle is at [tex]\((2, -8)\)[/tex], that gives us [tex]\(h = 2\)[/tex] and [tex]\(k = -8\)[/tex].
The radius [tex]\(r\)[/tex] is given as 11.
First, we plug in the values for [tex]\(h\)[/tex] and [tex]\(k\)[/tex] into the standard form:
[tex]\[
(x - 2)^2 + (y + 8)^2 = r^2
\][/tex]
Next, we need to calculate [tex]\(r^2\)[/tex], which is [tex]\(11^2\)[/tex]:
[tex]\[
r^2 = 11^2 = 121
\][/tex]
So the equation of the circle becomes:
[tex]\[
(x - 2)^2 + (y + 8)^2 = 121
\][/tex]
On examining the provided choices:
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]
We see that choice 2 matches our derived equation:
[tex]\[
(x - 2)^2 + (y + 8)^2 = 121
\][/tex]
Hence, the correct equation representing the circle with a center at [tex]\((2, -8)\)[/tex] and a radius of 11 is:
[tex]\[
(x - 2)^2 + (y + 8)^2 = 121
\][/tex]
