Determine the equation of the circle with center \( (-8,-5) \) containing the point \( (-11,4) \).

To find the equation of the circle, we start with the standard form of a circle's equation, which is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) is the center of the circle, and \( r \) is the radius. In this case, the center is \( (-8, -5) \). Next, we need to calculate the radius \( r \), which is the distance from the center \( (-8, -5) \) to the point \( (-11, 4) \). We can use the distance formula: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting \( (x_1, y_1) = (-8, -5) \) and \( (x_2, y_2) = (-11, 4) \): \[ r = \sqrt{((-11) - (-8))^2 + (4 - (-5))^2} \] \[ = \sqrt{(-11 + 8)^2 + (4 + 5)^2} \] \[ = \sqrt{(-3)^2 + (9)^2} \] \[ = \sqrt{9 + 81} \] \[ = \sqrt{90} \] \[ = 3\sqrt{10} \] Now that we have the radius \( r \), we can plug the center and the radius back into the circle's equation: \[ (x - (-8))^2 + (y - (-5))^2 = (3\sqrt{10})^2 \] This simplifies to: \[ (x + 8)^2 + (y + 5)^2 = 90 \] So, the equation of the circle is: \[ (x + 8)^2 + (y + 5)^2 = 90 \]