To determine the center and radius of the circle given by the equation [tex]\((x - 2)^2 + y^2 = 9\)[/tex], we can compare it to the standard form of the equation of a circle, which 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] represents the radius.
1. Identify the center [tex]\((h, k)\)[/tex]:
The given equation is [tex]\((x - 2)^2 + y^2 = 9\)[/tex]. From the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can see that:
- [tex]\(h\)[/tex] is the value that [tex]\(x\)[/tex] is subtracted from inside the parentheses. Here, [tex]\(x\)[/tex] is subtracted from 2, so [tex]\(h = 2\)[/tex].
- [tex]\(k\)[/tex] is the value that [tex]\(y\)[/tex] is subtracted from inside the parentheses. Here, [tex]\(y\)[/tex] is not subtracted from anything, so [tex]\(k = 0\)[/tex].
Therefore, the center of the circle is [tex]\((2, 0)\)[/tex].
2. Identify the radius [tex]\(r\)[/tex]:
The right-hand side of the equation is [tex]\(r^2\)[/tex]. In the given equation, this value is 9. To find the radius [tex]\(r\)[/tex], take the square root of 9:
[tex]\[
r = \sqrt{9} = 3
\][/tex]
Therefore, the radius of the circle is 3.
So, the answers are:
- The center of the circle is [tex]\((2, 0)\)[/tex].
- The radius of the circle is 3.
