2021 AMC 12A Problems/Problem 21

Problem

The five solutions to the equation $(z-1)(z^{2}+2z+4)(z^{2}+4z+6)=0$ may be written in the form $x_{k}+y_{k}i$ for $1\leq k\leq 5$ , where $x_{k}$ and $y_{k}$ are real. Let $\mathbb{E}$ be the unique ellipse that passes through the points $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}), (x_{4}, y_{4}),$ and $(x_{5}, y_{5})$ . The excentricity of $\mathbb{E}$ can be written in the form $\frac{m}{\sqrt{n}}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$ ?

$\textbf{(A) } 7\qquad\textbf{(B) } 9\qquad\textbf{(C) } 11\qquad\textbf{(D) } 13\qquad\textbf{(E) } 15\qquad$

Solution

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2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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2021 AMC 12A Problems/Problem 21

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