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**arze** A circle passes through the points A, B, and C which have coordinates $\displaystyle (0,3), (\sqrt{3},0), (-\sqrt{3},0)$ respectively.

Equation of circle $\displaystyle x^2+y^2-2y=3$

A line y=mx-3of variable gradient m, cuts the circle at L and M. Find the Cartesian equation of the locus of the midpoint of the line LM.

I found the values of x and y in terms of m.

$\displaystyle (\frac{4m+2\sqrt{m^2-3}}{m^2+1}, \frac{m^2+2m\sqrt{m^2-3}-3}{m^2+1})$ $\displaystyle (\frac{4m-2\sqrt{m^2-3}}{m^2+1}, \frac{m^2-2m\sqrt{m^2-3}-3}{m^2+1})$

Now what's next? I don't know hot to proceed from here. Any help is greatly appreciated! Thanks!