To determine the center and radius of the circle from its equation, let's examine the given equation carefully:
[tex]\[
(x - 3)^2 + (y + 2)^2 = 25
\][/tex]
This is the standard form of the equation of a circle, which is written as:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
From the standard form, we can directly identify the parameters:
1. Center of the Circle:
- The center [tex]\((h, k)\)[/tex] can be found directly from the terms [tex]\((x - h)\)[/tex] and [tex]\((y - k)\)[/tex] in the equation.
- In our equation, [tex]\((x - 3)\)[/tex] means [tex]\(h = 3\)[/tex], and [tex]\((y + 2)\)[/tex] means [tex]\(k = -2\)[/tex] (notice that [tex]\((y - k)\)[/tex] involves a negative sign, so if it's [tex]\((y + 2)\)[/tex], it means [tex]\(k = -2\)[/tex]).
- Therefore, the center of the circle is [tex]\((3, -2)\)[/tex].
2. Radius of the Circle:
- The radius [tex]\(r\)[/tex] can be found from the constant term on the right side of the equation.
- From the equation, [tex]\(r^2 = 25\)[/tex].
- Taking the square root of both sides, we get [tex]\(r = \sqrt{25} = 5\)[/tex].
So the final answers are:
The center of the circle is [tex]\((3, -2)\)[/tex].
The radius of the circle is [tex]\(5\)[/tex].
