<s>[INST] Find the locus of $|z-2i|=3|z+3|$ I got as far as 
$$|z-2i|=3|z+3| \Leftrightarrow \\ 
(\ldots) \Leftrightarrow \\
x^2-y^2+4y-4=9x^2+54x+81-9y^2 \Leftrightarrow \\
x^2-9x^2-y^2+9y^2+4y-4-54x-81=0\Leftrightarrow \\
-8x^2+8y^2+4y-85-54x=0 \Leftrightarrow \\
8y^2+4y-8x^2-54x-85=0\Leftrightarrow \\
y^2+\frac{1}{2}y-x^2-\frac{27}{4}x=\frac{85}{8}$$
Then I tried to complete the square:
$$y^2+\frac{1}{2}y = 0\Leftrightarrow (y+\frac{1}{4})^2-\frac{1}{16}$$
$$-x^2-\frac{27}{4}x=0 \Leftrightarrow -(x+\frac{27}{8})^2+\frac{729}{64} = 0 \Leftrightarrow (x+\frac{27}{8})^2-\frac{729}{64} = 0$$
And so the equation becomes:
$$(y+\frac{1}{4})^2-\frac{1}{16}+(x+\frac{27}{8})^2-\frac{729}{64}=\frac{85}{8} \Leftrightarrow \\
(y+\frac{1}{4})^2+(x+\frac{27}{8})^2=\frac{85}{8}+\frac{1}{16}+\frac{729}{64} $$
According to my book the center is $-\frac{27}{8}-\frac{1}{4}i$ which I got right but the radius is $\frac{\sqrt{117}}{8}$.
What went wrong?
 [/INST] Problem is way up at the top. You wrote $x^2 - y^2 + 4y-4$, when it should have been  $x^2 + y^2 - 4y+4$. Similarly, $9x^2 - 9y^2 + 54 x + 81$ should have been $9x^2 + 9y^2 + 54 x + 81$.
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