2020 AMC 12A Problems/Problem 15
Problem
In the complex plane, let be the set of solutions to
and let
be the set of solutions to
What is the greatest distance between a point of
and a point of
Solution
Realize that will create an equilateral triangle on the complex plane with the first point at
and two other points with equal magnitude at $1\pmi\sqrt{3}$ (Error compiling LaTeX. Unknown error_msg).
Also, realize that can be factored through grouping:
will create points at
and
Plotting the points and looking at the graph will make you realize that $1\pmi\sqrt{3}$ (Error compiling LaTeX. Unknown error_msg) and are the farthest apart and through Pythagorean Theorem, the answer is revealed to be
~lopkiloinm