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I'm trying to solve the following equation:

$ |\frac{1}{z}| = |z-1|$

My attempt was to make $z= x + iy$ but I get to a very complex equation $x^2+y^2 = (x^2 - 2x +1 +y^2)(x^4 +2x^2y^2 + y^4)$

I was wondering if there is a simpler way to do this. I also thought of making $|\frac{1}{z}| = |\frac{\overline{z}}{z\overline{z}}|$ but I don't feel it's gonna help me? Can someone give a hint? Thanks!

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1 Answer 1

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There are actually infinitely many $z$ which satisfy your equation. It's equivalent to $$ 1 = | z^2 - z|,$$ so you can find solutions by using the quadratic formula on $$ z^2 - z + c = 0,$$ where $c$ is any complex number with $|c| = 1$.

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