After using the midpoint formula to find the circle's center, the result is -3 - 6.5i, which does not match any of the provided options. This suggests an error in the question or the answer choices.
The student asks for the center of the circle with a diameter having endpoints at -2i and -6-11i. To find the center of the circle, we calculate the midpoint of the diameter by averaging the x-coordinates and the y-coordinates of the endpoints separately.
Midpoint formula: \( (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) \).
Let's apply the formula for the given points:
\( x_1 = 0 \) (since there is no real part in -2i)
\( y_1 = -2 \) (imaginary part of -2i)
\( x_2 = -6 \) (real part of -6-11i)
\( y_2 = -11 \) (imaginary part of -6-11i)
Now let's find the midpoint:
\( x_{mid} = (0 + (-6)) / 2 = -3 \)
\( y_{mid} = (-2 + (-11)) / 2 = -6.5 \)
So the midpoint, which is the center of the circle, is -3 - 6.5i. However, this is not one of the given options. Since none of the choices match, there may be a typo in the question or in the provided answer choices. The correct answer should be recalculated once the information is verified.