To determine the equation of a circle given its center and radius, we'll use the standard form of a circle's equation: [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex].
Here's the step-by-step process:
1. Identify the center of the circle [tex]$(h, k)$[/tex] and the radius [tex]\(r\)[/tex]:
- The center of the circle is given as [tex]\((3, 2)\)[/tex].
- The radius of the circle is given as [tex]\(5\)[/tex].
2. Substitute the given values into the standard form equation [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex]:
- Here, [tex]\(h = 3\)[/tex], [tex]\(k = 2\)[/tex], and [tex]\(r = 5\)[/tex].
3. Substitute [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r\)[/tex] into the equation:
- Replace [tex]\(h\)[/tex] with [tex]\(3\)[/tex], [tex]\(k\)[/tex] with [tex]\(2\)[/tex], and [tex]\(r\)[/tex] with [tex]\(5\)[/tex]:
[tex]\[
(x-3)^2 + (y-2)^2 = 5^2
\][/tex]
4. Simplify the equation:
- Calculate the square of the radius [tex]\(5\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
- So, the equation becomes:
[tex]\[
(x-3)^2 + (y-2)^2 = 25
\][/tex]
The equation of the circle with center [tex]\((3, 2)\)[/tex] and radius [tex]\(5\)[/tex] is [tex]\((x-3)^2 + (y-2)^2 = 25\)[/tex].
Therefore, the correct answer is:
A. [tex]\((x-3)^2 + (y-2)^2 = 25\)[/tex]
