<s>[INST] Consider the equation $|z + 3i|=3|z|$ for complex z and give a geometric description of the set S of all solutions. Writing $z$ in the form $a+ib$ and then rearranging gives $-8a^2-8b^2+6b+9=0$. The most promising form I could manage from this is $(b-\frac{3}{8})^2=(\frac{9}{8}-a)(\frac{9}{8}+a)$ but I still do not know how this looks like.
 [/INST] If $z=x+iy$ then 
\begin{align*}
|z+3i|&=3|z|&\;\iff\;&&x^2+(y+3)^2&=9(x^2+y^2)\\
      &     &\;\iff\;&&8x^2+8y^2-6y-9&=0\\       
      &     &\;\iff\;&& x^2+ y^2-\frac{3}{4}y-\frac{9}{8}&=0\\       
      &     &\;\iff\;&& x^2+ \left(y-\frac{3}{8}\right)^2&=\frac{81}{64}\\
\end{align*}
Therefore the set $S$ is a circle with center in $z_0=\frac{3}{8}i\;$ and radius $\frac{9}{8}$.
</s>