To determine the complete equation of the given circle, let's analyze it step by step.
1. Identify the given information:
- The standard form of the equation of a circle is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where:
- [tex]\((h, k)\)[/tex] is the center of the circle.
- [tex]\(r\)[/tex] is the radius of the circle.
2. Extract the center coordinates ( [tex]\(h\)[/tex], [tex]\(k\)[/tex] ) from the equation:
- The given equation is [tex]\((x - 2)^2 + (y + 2)^2 = r^2\)[/tex].
- Comparing this with the standard form, we see that:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = -2\)[/tex]
3. Determine the radius [tex]\(r\)[/tex]:
- The radius [tex]\(r\)[/tex] is not directly provided in the given equation.
- As a result, we cannot explicitly determine the value of [tex]\(r\)[/tex] from the given information alone. Hence, we denote the radius as unknown.
So, the equation of the circle with the given information is:
[tex]\[
(x - 2)^2 + (y + 2)^2 = r^2
\][/tex]
where [tex]\(r\)[/tex] remains unknown. Hence, the term [tex]\((x - 2)^2 + (y + 2)^2 = [?]\)[/tex] can be completed as follows:
[tex]\[
(x - 2)^2 + (y + 2)^2 = r^2
\][/tex]
