To determine the equation of a circle given its center and radius, we use the standard form of the equation of a circle:
[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.
For the given problem:
- The center of the circle is [tex]\((5, -3)\)[/tex]. Thus, [tex]\(h = 5\)[/tex] and [tex]\(k = -3\)[/tex].
- The radius of the circle is 4. Thus, [tex]\(r = 4\)[/tex].
We’ll substitute these values into the standard form equation:
1. Substitute [tex]\(h = 5\)[/tex] and [tex]\(k = -3\)[/tex] into the equation:
[tex]\[
(x - 5)^2 + (y - (-3))^2 = r^2
\][/tex]
2. Simplify the equation where necessary:
[tex]\[
(x - 5)^2 + (y + 3)^2 = r^2
\][/tex]
3. Calculate the value of [tex]\(r^2\)[/tex] (the square of the radius):
[tex]\[
r^2 = 4^2 = 16
\][/tex]
4. Substitute [tex]\(r^2 = 16\)[/tex] into the equation:
[tex]\[
(x - 5)^2 + (y + 3)^2 = 16
\][/tex]
So, the completed equation of the circle is:
[tex]\[
(x - 5)^2 + (y + 3)^2 = 16
\][/tex]
Therefore, the final answer with the blanks filled in is:
[tex]\[
(x - [5])^2 + (y - [-3])^2 = [16]
\][/tex]
