To determine the equation of a circle given its center and radius, let's refer to the standard form equation of a circle:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
Given:
- The center of the circle is [tex]\((4, 5)\)[/tex].
- The radius is [tex]\(3\)[/tex].
Now, we simply substitute [tex]\(h = 4\)[/tex], [tex]\(k = 5\)[/tex], and [tex]\(r = 3\)[/tex] into the standard form equation:
[tex]\[
(x - 4)^2 + (y - 5)^2 = 3^2
\][/tex]
Next, we know [tex]\(3^2\)[/tex] is equal to [tex]\(9\)[/tex], so we can rewrite the equation as:
[tex]\[
(x - 4)^2 + (y - 5)^2 = 9
\][/tex]
Thus, the equation of the circle is:
[tex]\[
(x - 4)^2 + (y - 5)^2 = 9
\][/tex]
Now, let's match this to the given options:
- A. [tex]\((x-4)^2+(y-5)^2=9\)[/tex]
- B. [tex]\((x-5)^2+(y-4)^2=9\)[/tex]
- C. [tex]\((x+4)^2+(y+5)^2=3\)[/tex]
- D. [tex]\(\left(x^2-4\right)+\left(y^2-5\right)=3^2\)[/tex]
From this, it is clear that option A matches our derived equation accurately. Therefore, the correct answer is:
[tex]\[
\boxed{(x-4)^2+(y-5)^2=9}
\][/tex]
Thus, the equation of the given circle is [tex]\((x-4)^2+(y-5)^2=9\)[/tex].
