Find an equation of the circle that has center \( (5,-1) \) and passes through \( (-1,4) \).

To find the equation of the circle with center \((5, -1)\) that passes through the point \((-1, 4)\), follow these steps: 1. **Standard Equation of a Circle**: The standard form of a circle's equation is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center and \(r\) is the radius. 2. **Identify the Center**: Given the center \((5, -1)\), substitute \(h = 5\) and \(k = -1\) into the equation: \[ (x - 5)^2 + (y + 1)^2 = r^2 \] 3. **Find the Radius**: The radius \(r\) is the distance between the center \((5, -1)\) and the point \((-1, 4)\). Use the distance formula: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] \[ r = \sqrt{(-1 - 5)^2 + (4 - (-1))^2} = \sqrt{(-6)^2 + (5)^2} = \sqrt{36 + 25} = \sqrt{61} \] Thus, \(r^2 = 61\). 4. **Final Equation**: Substitute \(r^2 = 61\) back into the standard equation: \[ (x - 5)^2 + (y + 1)^2 = 61 \] **Answer**: \[ (x - 5)^2 + (y + 1)^2 = 61 \]