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Math Help - image of a circle

  1. #1
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    image of a circle

    image of a circle-d.jpg

    can anybody please help me solve this one ? this is the last question i have ! thank you !!!
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  2. #2
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    If you put z = ce^{i\theta} (so that z=x+iy lies on the circle |z|=c), then

    w = \frac12(z+z^{-1}) = \frac12(ce^{i\theta} + c^{-1}e^{-i\theta}) = \frac12(c+c^{-1})\cos\theta + \frac i2(c-c^{-1})\sin\theta.

    So w lies on the ellipse \dfrac{x^2}{\bigl(\frac12(c+c^{-1})\bigr)^2} + \dfrac{y^2}{\bigl(\frac12(c-c^{-1})\bigr)^2}  = 1.
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  3. #3
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    Thank you a WHOLE lot
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