Question 21
The five solutions to the equation $$
\begin{aligned} &(z - 1)(z^2 + 2z + 4) \\ &\quad {}\cdot (z^2 + 4z + 6) = 0 \end{aligned}
$$ may be written in the form $x_k + y_k i$ for $1 \le k \le 5,$ where $x_k$ and $y_k$ are real. Let $E$ 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 eccentricity of $E$ can be written in the form $\sqrt{\tfrac{m}{n}},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$
(Recall that the eccentricity of an ellipse $E$ is the ratio $\tfrac{c}{a},$ where $2a$ is the length of the major axis of $E$ and $2c$ is the distance between its two foci.)
\begin{aligned} &(z - 1)(z^2 + 2z + 4) \\ &\quad {}\cdot (z^2 + 4z + 6) = 0 \end{aligned}
$$ may be written in the form $x_k + y_k i$ for $1 \le k \le 5,$ where $x_k$ and $y_k$ are real. Let $E$ 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 eccentricity of $E$ can be written in the form $\sqrt{\tfrac{m}{n}},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$
(Recall that the eccentricity of an ellipse $E$ is the ratio $\tfrac{c}{a},$ where $2a$ is the length of the major axis of $E$ and $2c$ is the distance between its two foci.)