Difference between revisions of "2021 AMC 12A Problems/Problem 21"

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==Problem==
 
==Problem==
The five solutions to the equation <cmath>(z-1)(z^{2}+2z+4)(z^{2}+4z+6</cmath> may be written in the form <math>x_{k}+y_{k}i</math> for <math>1\leq k\leq 5</math>, where <math>x_{k}</math> and <math>y_{k}</math> are real. Let <math>\mathbb{E}</math> be the unique ellipse that passes through the points <math>(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}), (x_{4}, y_{4}),</math> and <math>(x_{5}, y_{5})</math>. The excentricity of <math>\mathbb{E}</math> can be written in the form <math>\frac{m}{\sqrt{n}}</math> where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. What is <math>m+n</math>?
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The five solutions to the equation <cmath>(z-1)(z^{2}+2z+4)(z^{2}+4z+6)=0</cmath> may be written in the form <math>x_{k}+y_{k}i</math> for <math>1\leq k\leq 5</math>, where <math>x_{k}</math> and <math>y_{k}</math> are real. Let <math>\mathbb{E}</math> be the unique ellipse that passes through the points <math>(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}), (x_{4}, y_{4}),</math> and <math>(x_{5}, y_{5})</math>. The excentricity of <math>\mathbb{E}</math> can be written in the form <math>\frac{m}{\sqrt{n}}</math> where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. What is <math>m+n</math>?
  
 
<math>\textbf{(A) } 7\qquad\textbf{(B) } 9\qquad\textbf{(C) } 11\qquad\textbf{(D) } 13\qquad\textbf{(E) } 15\qquad</math>
 
<math>\textbf{(A) } 7\qquad\textbf{(B) } 9\qquad\textbf{(C) } 11\qquad\textbf{(D) } 13\qquad\textbf{(E) } 15\qquad</math>

Revision as of 15:07, 11 February 2021

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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See also

2021 AMC 12A (ProblemsAnswer KeyResources)
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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