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Complex Analysis: Locus Sketching

  1. Oct 4, 2008 #1
    1. The problem statement, all variables and given/known data

    Sketch the locus of |z-2i|=z+3 in C

    2. The attempt at a solution

    Let z=x+iy, then |z-i|=|x+iy-2i)|=|x+i(y-2)|=(x^2+(y-2)^2)^(1/2)=z+3

    The problem is that I can't tell what this means geometrically. Is it a spiral?
     
  2. jcsd
  3. Oct 4, 2008 #2

    tiny-tim

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    Hi altcmdesc! :smile:

    Hint: |z - 2i| is real, so z + 3 must be real, so … ? :wink:
     
  4. Oct 4, 2008 #3
    So it must be the real line?
     
  5. Oct 4, 2008 #4

    tiny-tim

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    It must be part of the real line.
     
  6. Oct 4, 2008 #5
    I'm thinking of this as the simpler problem z=|z-i| first, but I'm having a hard time believing this could be true for any z because of the triangle inequality.
     
  7. Oct 5, 2008 #6

    tiny-tim

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    That's right! :smile:

    But how about z+1=|z-i| ? :wink:
     
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