I am not sure you have have y + 3 as a factor. All points (x, -3) satisfy this equation, and they are not even inside the circle.

Here is one solution to (d). Let's lower the circle by 1 so that the center is at (0, 0). Suppose the middle of a chord LM has coordinates (x_{0}, y_{0}). Then the line from (0, 0) to (x_{0}, y_{0}) is perpendicular to LM, so the line LM has the equation xx_{0}+ yy_{0}= c for some constant c. (If this is unclear, please say so.) Since (x_{0}, y_{0}) ∈ LM, . Now, since (0, -4) ∈ line LM, . This is the required equation in x_{0}and y_{0}, only the locus has to lifted back up by 1.