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Greatest Distance in Complex Plane

Source: 2020 AMC 12A #15

January 31, 2020
AMCAMC 122020 AMC 12AAMC 12 A2020 AMCcomplex plane

Problem Statement

In the complex plane, let AA be the set of solutions to z3−8=0z^3 - 8 = 0 and let BB be the set of solutions to z3−8z2−8z+64=0z^3 - 8z^2 - 8z + 64 = 0. What is the greatest distance between a point of AA and a point of B?B?
<spanclass=′latex−bold′>(A)</span>23<spanclass=′latex−bold′>(B)</span>6<spanclass=′latex−bold′>(C)</span>9<spanclass=′latex−bold′>(D)</span>221<spanclass=′latex−bold′>(E)</span>9+3<span class='latex-bold'>(A) </span> 2\sqrt{3} \qquad <span class='latex-bold'>(B) </span> 6 \qquad <span class='latex-bold'>(C) </span> 9 \qquad <span class='latex-bold'>(D) </span> 2\sqrt{21} \qquad <span class='latex-bold'>(E) </span> 9 + \sqrt{3}
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