<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] If $|z+4|=3|z|$, and we let $z=x+\mathrm iy$, we get
$$|x+\mathrm i y + 4| = 3|x+\mathrm i y|$$
$$|(x+4)+\mathrm iy|=3|x+\mathrm i y|$$
$$\sqrt{(x+4)^2+y^2}=3\sqrt{x^2+y^2}$$
$$(x+4)^2+y^2 = 9(x^2+y^2)$$
$$x^2+8x+16+y^2=9x^2+9y^2$$
$$0=8x^2-8x+8y^2-16$$
$$0=x^2-x+y^2-2$$
$$0=\left(x-\frac{1}{2}\right)^2-\frac{1}{4}+y^2-2$$
$$\frac{9}{4} = \left(x-\frac{1}{2}\right)^2+y^2$$
This is a circle, centre $(1/2,0)$ with radius $3/2$, i.e.
$$\left|z-\frac{1}{2}\right|=\frac{3}{2}$$
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