Saturday, 23 April 2016

complex analysis - Describe the set whose points satisfy the following relation: |z21|<1



There is a hint which states to use polar coordinates, but I feel like that complicates the problem more. As far as trying it myself, I get lost very early on. If we take z=r(cosθ+isinθ), then we have



|r2(cos2θ+isin2θ)1|<1




But I have no idea how to find the modulus of this point with that extra 1 in there.


Answer



Hint



Taking the square, we get after simplifying



r2<2cos(2θ)



The set is inside the curve defined by its polar equation




r=2cos(2θ)


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