To determine which equation represents a circle with a center at [tex]\((2, -8)\)[/tex] and a radius of 11, we need to use the standard form of the equation of a circle. The standard form is given by:
[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 is [tex]\((2, -8)\)[/tex] and the radius is 11, we can substitute these values into the standard form of the equation.
1. Identify the center and radius:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = -8\)[/tex]
- [tex]\(r = 11\)[/tex]
2. Substitute these values into the standard form:
[tex]\[
(x - 2)^2 + (y + 8)^2 = 11^2
\][/tex]
3. Simplify the right-hand side of the equation:
[tex]\[
(x - 2)^2 + (y + 8)^2 = 121
\][/tex]
Therefore, the equation that represents 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]
Let's verify:
- The given options are:
- [tex]\((x - 8)^2 + (y + 2)^2 = 11\)[/tex]
- [tex]\((x - 2)^2 + (y + 8)^2 = 121\)[/tex]
- [tex]\((x + 2)^2 + (y - 8)^2 = 11\)[/tex]
- [tex]\((x + 8)^2 + (y - 2)^2 = 121\)[/tex]
Out of these options, the second option [tex]\((x - 2)^2 + (y + 8)^2 = 121\)[/tex] is the correct one.
