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Math Help - A problem with complex numbers and inequalities

  1. #1
    Junior Member Dark Sun's Avatar
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    A problem with complex numbers and inequalities

    W.t.s. there are complex numbers z satisfying |z-a|+|z+a|=2|c| iff |a|\leq |c|. Also, I need to find the smallest and largest values of z provided this condition is fulfilled.

    Step 1 is simple: Assume there are complex numbers z satisfying |z-a|+|z+a|=2|c|. Then, 2|c|=|z-a|+|z+a|=|a-z|+|z+a|\geq 2|a|,\mbox{ hence }|c|\geq |a|.

    So far for step 2 I have: Assume that the condition |a|\leq |c| is fullfilled. If z=-|c| is our smallest value, and z=|c| is our largest value, then we have 2|c|=2|z|\leq |z-a|+|z+a|, and I am now lacking equality.

    If I could show that \frac{|c|-a}{|c|+a}\geq 0, this would provide equality, however, there is still the matter of the complex number a, and since there is no order relation for complex numbers, this is proving difficult.

    I have also gotten |z-a|+|z+a|\leq |z|+|-a|+|z|+|a|=2|c|+2|a|\leq 4|c|, which is off by a factor of 2.

    I think my value for z is wrong.

    Any help would be greatly appreciated!
    Last edited by Dark Sun; January 12th 2010 at 11:27 PM.
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  2. #2
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    Quote Originally Posted by Dark Sun View Post
    W.t.s. there are complex numbers z satisfying |z-a|+|z+a|=2|c| iff |a|\leq |c|. Also, I need to find the smallest and largest values of z provided this condition is fulfilled.
    What do you mean by smallest and largest, if z is a complex number?

    If that may help you: |z-a|+|z+a|=2c is the equation of an ellipse with foci \pm a, long half-axis c and short half-axis \sqrt{c^2-|a|^2}.
    Last edited by Laurent; January 15th 2010 at 08:38 AM. Reason: Error in long half-axis: no factor 2
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  3. #3
    Junior Member Dark Sun's Avatar
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    I'm sorry, I meant the largest and smallest values of |z|.

    I will look into this ellipse, and report back.
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  4. #4
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    Quote Originally Posted by Dark Sun View Post
    I'm sorry, I meant the largest and smallest values of |z|.

    I will look into this ellipse, and report back.
    Then the extreme values are 0 and |c|. Triangular inequality from 2z=(z+a)+(z-a), and considering z=|c|\frac{a}{|a|} should do it.
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