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 \( (5, -1) \) and passing through the point \( (-1, 4) \), we start by calculating the radius. The radius \( r \) can be found using the distance formula between the center and the point on the circle. The distance formula is given by: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] In this case, \( (x_1, y_1) = (5, -1) \) and \( (x_2, y_2) = (-1, 4) \). Calculating the differences: \[ x_2 - x_1 = -1 - 5 = -6 \] \[ y_2 - y_1 = 4 - (-1) = 4 + 1 = 5 \] Now we can substitute these into the distance formula: \[ r = \sqrt{(-6)^2 + (5)^2} = \sqrt{36 + 25} = \sqrt{61} \] Now that we have the radius, we can write the equation of the circle in the standard form: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center of the circle. Substituting in our values: \[ (x - 5)^2 + (y + 1)^2 = (\sqrt{61})^2 \] Thus, the equation of the circle is: \[ (x - 5)^2 + (y + 1)^2 = 61 \]