Score: \( 5 / 6 \) Penalty:none Question Determine the equation of the circle with center \( (-8,8) \) containing the point \( (4,24) \) Answer Attempt 1 out of 2

To find the equation of a circle, we start with the standard form: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center of the circle, and \(r\) is the radius. Given the center \((-8, 8)\), we have \(h = -8\) and \(k = 8\). Next, we need to find the radius \(r\) by determining the distance from the center to the point \((4, 24)\). Using the distance formula: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(4 - (-8))^2 + (24 - 8)^2} \] Calculating this, we get: \[ r = \sqrt{(4 + 8)^2 + (24 - 8)^2} = \sqrt{(12)^2 + (16)^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \] Now, substituting the center \((-8, 8)\) and radius \(20\) into the equation of the circle gives: \[ (x + 8)^2 + (y - 8)^2 = 20^2 \] Finally, we write the equation: \[ (x + 8)^2 + (y - 8)^2 = 400 \]