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Math Help - Transformations in the Compex Plane

  1. #1
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    Transformations in the Compex Plane

    For the transformation w=1/(z-2), where z=x+iy and w=u+iv,
    show that the straight line 2x+y=5 is transformed to a circle with centre (1,-0.5) and radius 0.5sqrt5.

    I can get to the stage where I equate the real and imaginary parts, but I can't see how to use the information about the original image, i.e the straight line 2x+y=5.

    Any help would be greatly appreciated, thanks.
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  2. #2
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    Quote Originally Posted by dbatten View Post
    For the transformation w=1/(z-2), where z=x+iy and w=u+iv,
    show that the straight line 2x+y=5 is transformed to a circle with centre (1,-0.5) and radius 0.5sqrt5.

    I can get to the stage where I equate the real and imaginary parts, but I can't see how to use the information about the original image, i.e the straight line 2x+y=5.
    Given w = 1/(z–2), solve that for z to get z = \frac1w+2 = \frac{u-iv}{u^2+v^2}+2. Take real and imaginary parts: x = \frac u{u^2+v^2}+2, y = \frac{-v}{u^2+v^2}. Now substitute those values of x and y into the equation 2x+y=5, and you get the equation for u and v.
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