To find the equation of a circle given its center and radius, we use the general formula for the equation of a circle:
[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.
Let's break down the components for the given circle:
1. The center of the circle is [tex]\((3, 2)\)[/tex], so [tex]\(h = 3\)[/tex] and [tex]\(k = 2\)[/tex].
2. The radius of the circle is [tex]\(5\)[/tex], so [tex]\(r = 5\)[/tex].
Substitute these values into the general formula:
[tex]\[
(x - 3)^2 + (y - 2)^2 = 5^2
\][/tex]
Next, simplify the right side of the equation:
[tex]\[
(x - 3)^2 + (y - 2)^2 = 25
\][/tex]
Now we just need to find the correct option that matches this equation. Let’s review the options:
A. [tex]\((x^2 - 3) + (y^2 - 2) = 5^2\)[/tex]
This is not correct because the equation is not properly formatted to match the standard circle equation. Also, it does not equal [tex]\(25\)[/tex] when simplified.
B. [tex]\((x + 3)^2 + (y + 2)^2 = 5\)[/tex]
This is not correct because the signs do not match the required [tex]\((x - 3)\)[/tex] and [tex]\((y - 2)\)[/tex], and the right side should be [tex]\(25\)[/tex], not [tex]\(5\)[/tex].
C. [tex]\((x - 2)^2 + (y - 3)^2 = 25\)[/tex]
This is not correct because the values for [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are swapped.
D. [tex]\((x - 3)^2 + (y - 2)^2 = 25\)[/tex]
This is correct because it matches our derived equation exactly.
Therefore, the correct answer is:
D. [tex]\((x - 3)^2 + (y - 2)^2 = 25\)[/tex]
