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$z$ is a complex number. What does $2|z+3i|=|z-i|$ represent?

Putting $z=x+iy$, I get a circle with radius $\frac83$. But the equation of circle is $|z-z_1|=r$.

How to intuitively see that $2|z+3i|=|z-i|$ represents a circle?

aarbee
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    This is a disguise of the circles of Apollonius. I haven't tried to figure out an intuitive explanation for this, but this keyword will certainly be helpful for googling your question. – Sangchul Lee Dec 16 '20 at 06:50
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    Once you square and simplify, the coefficient of $x^2$ equals the coefficient of $y^2$, and there is no $xy$, so it will be a circle – Empy2 Dec 16 '20 at 06:57
  • If you know that inversion preserves circles, then it suffices to check that the image of the locus under the transformation $w=\frac{1}{z-i}$ is also a circle, which is much easier to check. – Sangchul Lee Dec 16 '20 at 07:42
  • @SangchulLee That's an interesting line of thinking. Mind developing that into an answer? Thanks. – aarbee Dec 16 '20 at 07:43
  • Since this posting is locked, I added my answer to the linked post, see this. – Sangchul Lee Dec 16 '20 at 08:24

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That is the Circles of Apollonius. In fact, if $|z - a| = k |z - b|$, we have $$ \left |z - \frac{a - b k^2}{1 - k^2}\right | = \frac{k|a - b|}{|1 - k^2|}, $$ Thus it is a circle. See this post.

FFjet
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