Hi everyone. The full question is as follows:

A circle passes through $\displaystyle A(0,3), B(\sqrt{3},0), C(-\sqrt{3},0)$. 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 $\displaystyle y=mx-3$ of variable gradient $\displaystyle m$, cuts the circle ABC in two points $\displaystyle L$ and $\displaystyle M$ 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: $\displaystyle x^{2}+y^{2}-2y=3$

For (b): $\displaystyle \frac{4\pi}{3}$

For (c): $\displaystyle x^{2}+y^{2}-\sqrt{3}x-3y=0$

I know that the answer to (d) is $\displaystyle (x^{2}+y^{2}+2y-3)(y+3)=0$. I also know that the line $\displaystyle y=mx-3$ "rotates" around the point $\displaystyle (0,-3)$

Thanks in advance.