<s>[INST] Displaying $ |z+4|=3|z|$ in an Argand diagram Identify in an Argand diagram the points corresponding to:
$$|z+4|=3|z|$$
I am unsure how to approach this question and would appreciate any help anyone can provide with the answer and the intuition behind the method. Thanks.
 [/INST] Hint: By substituting $z=x+iy$ we get
\begin{align}
|x+4+yi|=3|x+yi|
& \implies \sqrt{(x+4)^2+y^2}=3\sqrt{x^2+y^2} \\
& \implies (x + 4)^2 + y^2 = 9(x^2 + y^2) \\
& \implies (x + 4)^2 - 9x^2 = 8y^2 \\
& \implies y = \pm \sqrt{\frac{(x + 4)^2 - 9x^2}{8}}
=\pm \frac{\sqrt{-8x^2 + 16x + 16}}{2 \sqrt{2}}
\end{align}
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