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