To identify the radius and the center of the circle given its equation [tex]\((x - 5)^2 + y^2 = 81\)[/tex], we can proceed with the following steps:
### Step-by-Step Solution:
1. Recognize the standard form of a circle's equation:
The standard form of a circle's equation is [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] represents the radius.
2. Compare the given equation with the standard form:
The given equation is [tex]\((x - 5)^2 + y^2 = 81\)[/tex]. By comparing it with the general form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can identify the values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r^2\)[/tex]:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = 0\)[/tex]
- [tex]\(r^2 = 81\)[/tex]
3. Identify the center of the circle [tex]\((h, k)\)[/tex]:
Using the values identified above:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = 0\)[/tex]
Therefore, the center of the circle is at [tex]\((5, 0)\)[/tex].
4. Determine the radius [tex]\(r\)[/tex]:
Since [tex]\(r^2 = 81\)[/tex], we take the square root of both sides to find the radius:
[tex]\[
r = \sqrt{81} = 9
\][/tex]
### Final Answer:
- The radius of the circle is [tex]\(9\)[/tex] units.
- The center of the circle is at [tex]\((5, 0)\)[/tex].
Thus:
- The radius of the circle is [tex]\(9\)[/tex] units.
- The center of the circle is at [tex]\((5, 0)\)[/tex].
