Question

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 $1 \le k \le 5,$ 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.)

$7$

$9$

$11$

$13$

$15$

Solution:

The roots are $z = 1,$ $z = -1 \pm i\sqrt3,$ and $z = -2 \pm i\sqrt2,$ giving the points $(1, 0),$ $(-1, \pm\sqrt3),$ and $(-2, \pm\sqrt2).$ By symmetry about the $x$ -axis, the ellipse has the form $Ax^2 + Cy^2 + Dx + F = 0.$

Substituting the points yields $5x^2 + 6y^2 + 9x - 14 = 0.$ Completing the square gives $5\left(x + \tfrac{9}{10}\right)^2 + 6y^2 = \tfrac{361}{20},$ so $a^2 = \tfrac{361}{100}$ (along $x$ ) and $b^2 = \tfrac{361}{120}.$ Then $e^2 = 1 - \dfrac{b^2}{a^2} = 1 - \dfrac{100}{120} = \dfrac16,$ so $e = \sqrt{\tfrac16}.$

With $m = 1$ and $n = 6,$ we get $m + n = 7.$

Thus, the correct answer is A.

2021 AMC 12A Spring Problem 21

Solution:

Problem 21 in Other Years