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Math Help - Complex numbers and loci

  1. #1
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    Complex numbers and loci

    If the real part of (z+1)/(z-1) is zero, find the locus of points representing z in the complex plane.

    The answer in the book is x^2 + y^2 = 1 ie. a circle, but i dont know how to solve it
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  2. #2
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    \displaystyle \frac{z+1}{z-1} = \frac{x+1 + iy}{x-1 + iy}

    \displaystyle = \frac{(x+1+iy)(x-1-iy)}{(x-1+iy)(x-1-iy)}

    \displaystyle = \frac{(x+1+iy)(x-1-iy)}{(x-1)^2 + y^2}

    \displaystyle = \frac{x^2 + y^2 - 1 - 2iy}{(x-1)^2 + y^2}

    \displaystyle = \frac{x^2 + y^2 - 1}{(x - 1)^2 + y^2} + i\left[\frac{-2y}{(x - 1)^2 + y^2}\right].


    You know that the real part is 0, so \displaystyle \frac{x^2 + y^2 - 1}{(x-1)^2 + y^2} = 0

    \displaystyle x^2 + y^2 - 1 = 0

    \displaystyle x^2 + y^2 = 1.
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