To find the equation of a circle with a given center and radius, we use the standard form of the equation of a circle, which is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the coordinates of the center of the circle, and [tex]\(r\)[/tex] represents the radius.
Given:
- The center of the circle is [tex]\((5, -3)\)[/tex]; therefore, [tex]\(h = 5\)[/tex] and [tex]\(k = -3\)[/tex].
- The radius of the circle is [tex]\(4\)[/tex]; thus, [tex]\(r = 4\)[/tex].
Now, let's substitute these values into the standard form of the equation.
1. Substitute [tex]\(h = 5\)[/tex]:
[tex]\[
(x - 5)^2
\][/tex]
2. Substitute [tex]\(k = -3\)[/tex]:
[tex]\[
(y - (-3))^2 = (y + 3)^2
\][/tex]
3. Calculate [tex]\(r^2\)[/tex]:
[tex]\[
r^2 = 4^2 = 16
\][/tex]
Therefore, the complete equation of the circle is:
[tex]\[
(x - 5)^2 + (y + 3)^2 = 16
\][/tex]
