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Complex numbers max magnitude system

2175ExpertComplex NumbersAlgebra

February 2017 · 2017

Let $a$ and $b$ be complex numbers satisfying the two equations $$ \begin{aligned} a^{3}-3 a b^{2} & =36 \\ b^{3}-3 b a^{2} & =28 i \end{aligned} $$ Let $M$ be the maximum possible magnitude of $a$. Find all $a$ such that $|a|=M$.
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