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If z is a complex number satisfying the equation $|z-(1+i)|^2 =2$ and m=2/z ,then what is the locus traced by m in the complex plane .

I know z is a locus of circle with centre (1,1) and radius $\sqrt 2$.

But not getting any idea to solve for m


answer is given as x-y-1=0

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  • $\begingroup$ Inversions in circles should be useful, because the map $f(z)=2/z$ is an inversion in the circle of radius $\sqrt{2}$ centered on the origin. $\endgroup$ – Lee Mosher Apr 20 '17 at 12:22
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Write $z=\frac 2m$ and substitute this into the locus given so you have $$\left|\frac{2-(1+i)m}{m}\right|^2=2$$ So $$|2-(m(1+i)|^2=2|m|^2$$

Now write $m=x+iy$ and do some simplification, and you will end up with $y=x-1$ as desired.

Shall I leave this to you?

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