To determine the center and radius of the circle given by the equation [tex]\((x-3)^2 + (y+2)^2 = 25\)[/tex], we follow these steps:
1. Recognize the general form of a circle's equation:
[tex]\[
(x-h)^2 + (y-k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] is the center of the circle, and [tex]\(r\)[/tex] is the radius.
2. Compare the given equation [tex]\((x-3)^2 + (y+2)^2 = 25\)[/tex] with the general form.
3. Identify the values for [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r^2\)[/tex]:
- The term [tex]\((x-3)^2\)[/tex] indicates that [tex]\(h = 3\)[/tex].
- The term [tex]\((y+2)^2\)[/tex] indicates that [tex]\(k = -2\)[/tex].
- The right hand side [tex]\(25\)[/tex] indicates that [tex]\(r^2 = 25\)[/tex].
4. Calculate the radius [tex]\(r\)[/tex] by taking the square root of [tex]\(r^2\)[/tex]:
[tex]\[
r = \sqrt{25} = 5
\][/tex]
Therefore, the center of the circle is [tex]\((3, -2)\)[/tex] and the radius is [tex]\(5\)[/tex].
The completed answers are:
The center of the circle is [tex]\((3, -2)\)[/tex].
The radius of the circle is [tex]\(5\)[/tex].
