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Doubt about the solutions of the equation aZ + bz + c = 0

  1. Nov 10, 2014 #1
    Dear all, can you please help me show why the 2nd scenario below has no solution......

    Theorem 3.3.1 Suppose that a and b are not both zero. Then the equation ## aZ + b\bar Z + c = 0 ##
    has
    a, b and c are complex numbers. Z is the unknown complex number.

    (1) a unique solution if and only if ## |a| != |b| ##;
    (2) no solution if and only if |a| = |b| and ## b\bar c != c\bar a ##;
    (3) a line of solutions if and only if |a| = |b| and ## b\bar c = c\bar a ##.

    Answer:

    (1) a unique solution

    In the equation ## aZ + b\bar Z + c = 0 ##
    a,b and c are complex numbers.

    for |a| = |b|, say ##a = f + ig##
    then |a| = ## \sqrt{f^2 + g^2} ##

    negative f and g will be equal to positive because of the square root

    so b can be ## f + ig, f - ig, -f + ig, -f -ig, g + if, g - if, -g + if, -g - if ##

    if |a| and |b| are equal i.e. when a and b are any of the complex numbers above then there may be repeated solutions for z.

    if ##z = z1 + iz2 ## then for two different a and b in the above list z1 will same and z2 will be same.

    (2) no solution

    if |a| = |b| then ONLY repeated solutions will occur and a and b are only one of the eight complex numbers given, c is any complex number???

    what does the condition ## b\bar c != c\bar a## do?


    (3) line of solutions
    (3) what does the condition ## b\bar c = c\bar a## do?
     
  2. jcsd
  3. Nov 12, 2014 #2

    chiro

    User Avatar
    Science Advisor

    Hey PcumP_Ravenclaw.

    The easiest way to do this (since it is a linear problem) is just set a = x + yi, b = d + ei, c = o + pi and Z = q + ri and get the solutions for the real and imaginary parts to equal 0.

    If you do this, then it will be easier to answer your own questions.
     
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