To identify the radius and the center of the circle given by the equation [tex]\((x-5)^2 + y^2 = 81\)[/tex], follow these steps:
### Step-by-Step Solution:
1. Understand the general form of the 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] is the center of the circle and [tex]\(r\)[/tex] is the radius.
2. Compare the given equation with the general form:
The given equation is [tex]\((x-5)^2 + y^2 = 81\)[/tex]. Compare this with the standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex].
3. Identify the center from [tex]\((x-h)^2 + (y-k)^2 = 81\)[/tex]:
- The term [tex]\((x-5)^2\)[/tex] indicates that [tex]\(h = 5\)[/tex].
- The term [tex]\(y^2\)[/tex] can be written as [tex]\((y - 0)^2\)[/tex], which indicates that [tex]\(k = 0\)[/tex].
Therefore, the center of the circle is [tex]\((h, k) = (5, 0)\)[/tex].
4. Identify the radius from [tex]\((x-h)^2 + (y-k)^2 = 81\)[/tex]:
- The right side of the equation [tex]\((x-5)^2 + y^2 = 81\)[/tex] is equal to [tex]\(r^2\)[/tex].
- Therefore, [tex]\(r^2 = 81\)[/tex].
- To find the radius [tex]\(r\)[/tex], take the square root of both sides: [tex]\(r = \sqrt{81}\)[/tex].
[tex]\[
r = 9
\][/tex]
5. Summarize the results:
- The radius of the circle is [tex]\(9\)[/tex] units.
- The center of the circle is at [tex]\((5, 0)\)[/tex].
### Final Answer:
The radius of the circle is [tex]\(9\)[/tex] units.
The center of the circle is at [tex]\((5, 0)\)[/tex].
