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Absolute values

  1. Sep 20, 2009 #1
    1. The problem statement, all variables and given/known data
    So I've got two problems I'm struggling a bit with. One of them I've solved (I think), but I'm definitely not sure. The other one is bugging me a bit. Anyway:

    i] Determine all z∈C so that |z - 1| = 5 and |z - 4| = 4

    ii] Determine all z∈C so that |4 - z2| = z


    2. Relevant equations



    3. The attempt at a solution
    i] I say that z = x+yi as a starting point. From there:

    |x + yi -1| = 5
    √( (x - 1)2 + y2 ) = 5
    x2 + 1 -2x +y2 = 25

    |x + yi -4| = 4
    √( (x-4)2 + y2 ) = 4
    x2 + 16 - 8x + y2 = 16

    y2 = 8x - x2

    Inserting this in the first equation:

    x2 + 1 - 2x + 8x - x2 = 25

    6x + 1 = 25

    x = 4

    and then y2 = 32 - 16 = 16, y = ± 4

    So I get z = 4±4i

    I think this should be correct, but I'm a bit.. unsure.


    ii] I've gotten so far that I've looked at the exercise and realised that the absolute value of someting is always a real number, which means if z = x+yi, then y=0. But from here I'm unsure on how to proceed.

    How on earth am I supposed to solve this? I'm feeling.. lost.
     
  2. jcsd
  3. Sep 20, 2009 #2

    ehild

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    i] is correct.

    ii] You are right, z is real. How do you define the absolute value of a real number?

    ehild
     
  4. Sep 20, 2009 #3
    The only thing I can think of right now (it's.. late) is:

    |x| = x if x>0
    |x| = 0 if x=0
    |x| = -x if x<0

    Is this what you mean?
     
  5. Sep 21, 2009 #4
    Never mind, I had a insight today during my lecture and suddenly it was all very, very clear and the answers are something like ±(1 + √17)/2

    Thanks though!
     
  6. Sep 21, 2009 #5

    ehild

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    Almost good! Do not forget that z can not be negative as it is equal to an absolute value. You had two second order equations, with 4 roots altogether, but only the positive roots are valid. (±1 + √17)/2

    ehild
     
  7. Sep 23, 2009 #6
    Yeah, sorry, I put the plus/minus sign wrong :) I figured that out and even checked if they were in the right intervals and such.
     
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