Transform complex equation into Cartesian coordinate Here is the problem
$$$$
Show the following equation is an ellipse
$$|z-1|=3-|z-2|$$
and I tried to solve it...
Square both sides,
$$(|z-1|)^2=(3-|z-2|)^2 \\
x^2+y^2-2x+1=9-6|z-2|+x^2+y^2-4x+4$$
Rearrange them...
$$6|z-2|=12-2x \\
3|z-2|=6-x$$
Square them again, and here is the problem i met...
$$(3|z-2|)^2=(6-x)^2 \\
9(x^2+y^2-4x+4)=36-12x+x^2 \\
$$
36 in both sides is going to be cancelled out, it seems like I must did something wrong. I have spent over 2 hours on this but still get the same result..please help!

So you get $8x^2+9y^2-24x=0$, which is $8\Big(x^2-3x+\frac{9}{4}\Big)-18+9y^2=0$, which is 
$$8\Big(x-\frac{3}{2}\Big)^2+9y^2 = 18$$
which is an ellipse.
