Real parameter given complex variable modulus

  1. 1. The problem statement, all variables and given/known data
    Suppose z is complex number with |z| = 1 (also assume that z is not 1 + 0*i)
    Let z = a + b*i where a and b are real numbers.

    Find a real parameter, t, such that

    z = (t-i)/(t+i), where i = sqrt(-1)


    2. Relevant equations


    |z| = sqrt(a^2+b^2)



    3. The attempt at a solution
    Since we are given |z| = 1, this implies that a^2 + b^2 = (a+b*i)*(a-b*i) = 1.
    Also, (t-i)/(t+i) can be expressed as [1/(t^2+1)]*[(t^2-1)+(-2*t)*i].

    Despite my efforts, I end up at two deadends: I reduce an equation to a tautology like 0 = 0 or I reduce to an expression for t that involves i (thereby making t imaginary instead of real).

    Any help is greatly appreciated.
     
  2. jcsd
  3. hunt_mat

    hunt_mat 1,613
    Homework Helper

    First off find c(t),d(t) such that:
    [tex]
    c(t)+d(t)i=\frac{t-i}{t+i}
    [/tex]
    Then use the fact that c(t)^{2}+d(t)^{2}=1
     
  4. SammyS

    SammyS 9,052
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Write

    [tex]\frac{1}{t^2+1}\,\left[(t^2-1)+(-2t)i\right][/tex]

    as

    [tex]\frac{(t^2-1)}{t^2+1}-\frac{2t}{t^2+1}\,i\right][/tex]
     
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