Hi everyone. The full question is as follows:

A circle passes through . Find:

(a)The equation of the circle

(b) the length of the minor arc BC

(c) the equation of the circle on AB as diameter.

(d) A Line of variable gradient , cuts the circle ABC in two points and Find in Cartesian form the equation of the locus of the midpoint of LM.

I have answered A through C, but (d) eludes me, and I can't even begin.

The answer for (a) is:

For (b):

For (c):

I know that the answer to (d) is . I also know that the line "rotates" around the point

Thanks in advance.