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Would be great to get your help in finding the locus of this complex number $z$: $|z-z_1|+\sin \alpha|z-z_2|=\sin \theta$

From this question I proceed to a refined one-
What would $$|z-z_1|+2|z-z_2|=k$$
represent?

DeepK
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  • I substituted z and expanded taking squares of both sides but it turned into an ugly expression with a square root term which i cannot get rid of... – DeepK May 06 '14 at 18:48
  • Just to be clear, you seem to be assuming that $a$ and $b$ are real. Are you sure about that? – mweiss May 06 '14 at 20:56
  • @mweiss no, sorry it was a blunder on my part... i would immediately correct it... – DeepK May 06 '14 at 21:12

1 Answers1

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Here's a hint. If $\alpha = \pi / 2$, then this equation says that the sum of the distances from $z$ to $z_1$ and $z_2$ is a fixed constant -- i.e. $z$ is on an ellipse with foci at $z_1$ and $z_2$. What happens if $\alpha=0$ or $\alpha = \pi$? Can you generalize?

mweiss
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  • Thanks... this intuitive approach is what I was looking for. I can see it would represent a circle, ellipse and hyperbola for 0, 1 and -1 as values of $\sin\alpha$ but what about the intermediary values of $\sin\alpha$? – DeepK May 07 '14 at 08:45