Question #138065
prove that image of
the vertical line x=k with k is not =0 to the circle mode of w-1/2k = mode of 1/2k
1
Expert's answer
2020-10-13T18:55:22-0400

z1z.z\rightarrow \frac{1}{z} . Hence (x+iy)1x+iy=xiyx2+y2(x+iy)\rightarrow \frac{1}{x+iy}=\frac{x-iy}{x^2+y^2} . Let u=xx2+y2,v=yx2+y2.u=\frac{x}{x^2+y^2}, v= \frac{-y}{x^2+y^2}. Then (u12k)2+v2=u2+v2uk+14k2=x2+y2(x2+y2)2uk+14k2(u-\frac{1}{2k})^2+ v^2= u^2+v^2-\frac{u}{k} +\frac{1}{4k^2} = \frac{x^2+y^2}{(x^2+y^2)^2}-\frac{u}{k}+\frac{1}{4k^2} =1x2+y2uk+14k2=\frac{1}{x^2+y^2}-\frac{u}{k}+\frac{1}{4k^2} But here x=kx=k is the given line. Hence u=kx2+y2uk=1x2+y2u=\frac{k}{x^2+y^2}\Rightarrow \frac{u}{k}=\frac{1}{x^2+y^2} . Hence RHS= 14k2\frac{1}{4k^2}. Let w=u+iv.w=u+iv. Then w12k=(u12k)2+v2=12k.|w-\frac{1}{2k}|=\sqrt{(u-\frac{1}{2k})^2+v^2}= \frac{1}{2k .}

Hence the image of the line is the circle.


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