Every circle has form |z-a|=k|z-b|

  • Thread starter Grothard
  • Start date
  • #1
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We can express any circle in the complex plane as |z-a|=k|z-b| where a and b are distinct complex numbers, k > 0 and [itex]k \not= 1.[/itex]

Is there an elegant way of showing this fundamental property of the complex plane to be true?
 

Answers and Replies

  • #2
mathman
Science Advisor
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It doesn't look elegant, but squaring both sides leads to a straightforward proof.
 
  • #3
AlephZero
Science Advisor
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Google for "circle of Apollonius". This geometrical result was known centuries before complex numbers and analytic geometry were invented.
 

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