To determine the equation of a circle given its center and radius, we use the standard form of the equation of a circle:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] represents the center of the circle and [tex]\(r\)[/tex] is the radius.
Given the center [tex]\((h, k)\)[/tex] of the circle is [tex]\((7, 8)\)[/tex] and the radius [tex]\(r\)[/tex] is 11, we can substitute these values into the standard equation.
First, identify the values from the given information:
- [tex]\(h = 7\)[/tex]
- [tex]\(k = 8\)[/tex]
- [tex]\(r = 11\)[/tex]
Next, plug these values into the standard equation:
[tex]\[
(x - 7)^2 + (y - 8)^2 = 11^2
\][/tex]
Now, calculate [tex]\(11^2\)[/tex]:
[tex]\[
11^2 = 121
\][/tex]
So the equation becomes:
[tex]\[
(x - 7)^2 + (y - 8)^2 = 121
\][/tex]
Thus, the correct equation for the circle is:
[tex]\[
(x - 7)^2 + (y - 8)^2 = 121
\][/tex]
Out of the provided options:
1. [tex]\((x-7)^2+(y-8)^2=121\)[/tex]
2. [tex]\((x-7)^2+(y-8)^2=11\)[/tex]
3. [tex]\((x+7)^2+(y+8)^2=121\)[/tex]
4. [tex]\((x+7)^2+(y+8)^2=11\)[/tex]
The correct answer is:
[tex]\[
(x-7)^2+(y-8)^2=121
\][/tex]
