Given the equation of the circle:
[tex]\[
(x - 11)^2 + (y - 15)^2 = 100
\][/tex]
We need to determine the center and the radius of the circle.
### Step-by-Step Solution:
1. Identify the general form of the circle's equation:
The general form of a circle's equation is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the center of the circle and [tex]\(r\)[/tex] is the radius.
2. Compare the given equation with the general form:
Looking at the given equation:
[tex]\[
(x - 11)^2 + (y - 15)^2 = 100
\][/tex]
and comparing it with the general form, we can identify the following:
- [tex]\(h = 11\)[/tex]
- [tex]\(k = 15\)[/tex]
So, the center of the circle, [tex]\((h, k)\)[/tex], is [tex]\((11, 15)\)[/tex].
3. Determine the radius:
In the general form of the equation, [tex]\(r^2\)[/tex] corresponds to the right-hand side of the equation.
[tex]\[
r^2 = 100
\][/tex]
To find the radius, [tex]\(r\)[/tex], we take the square root of both sides:
[tex]\[
r = \sqrt{100} = 10
\][/tex]
### Final Answer:
- The center has coordinates [tex]\((11, 15)\)[/tex].
- The radius is [tex]\(10\)[/tex] units long.
