2) \( (x+1)^{2}+(y-5)^{2}=4 \)

The equation given is \( (x+1)^{2}+(y-5)^{2}=4 \). This represents a circle in the Cartesian coordinate system. ### Known Conditions: 1. The center of the circle is at the point \((-1, 5)\). 2. The radius of the circle is \( r = \sqrt{4} = 2 \). ### Step-by-Step Explanation: 1. **Identify the center**: The center of the circle can be found from the standard form of the circle's equation, which is \( (x-h)^{2} + (y-k)^{2} = r^{2} \), where \((h, k)\) is the center and \(r\) is the radius. - Here, \(h = -1\) and \(k = 5\), so the center is \((-1, 5)\). 2. **Calculate the radius**: The radius is the square root of the right side of the equation. - Since \(4\) is the right side, the radius \(r = \sqrt{4} = 2\). ### Conclusion: The circle has a center at \((-1, 5)\) and a radius of \(2\). If you need further analysis or specific points on the circle, please let me know!