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Math Help - Complex Number

  1. #1
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    Complex Number

    Let  z_1, z_2  be nonzero complex numbers such that

    z_1^2 + z_2^2 = \sqrt{2}z_1 z_2 .

    Show that | z_1 | = | z_2 | and find Arg{(\frac{z_1}{z_2})}.

    I've tried substituting z = x+yi and z = re^{i\theta} to prove this but it didnt work out. I assume there is a geometric or inequality approach to this question?
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  2. #2
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    Quote Originally Posted by shinn View Post
    Let  z_1, z_2  be nonzero complex numbers such that

    z_1^2 + z_2^2 = \sqrt{2}z_1 z_2 .

    Show that | z_1 | = | z_2 | and find Arg{(\frac{z_1}{z_2})}.

    I've tried substituting z = x+yi and z = re^{i\theta} to prove this but it didnt work out. I assume there is a geometric or inequality approach to this question?
    In the equation z_1^2 + z_2^2 = \sqrt{2}z_1 z_2 , divide through by z_2^{\,2}, and let w = z_1/z_2. You will get a quadratic equation for w.
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