e 0 \). However, since the equation is symmetric in \((x,y)\) under this proportionality and no restriction on sign, the solution set is:

February 22, 2026

$e 0 \). However, since the equation is symmetric in \((x,y)\) under this proportionality and no restriction on sign, the solution set is:$

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Similarly for \(x = -\sqrt{5} y\):
\( rac{-\sqrt{5}+1}{-\sqrt{5}-1} = rac{\sqrt{5}-1}{\sqrt{5}+1} = rac{(\sqrt{5}-1)^2}{5 - 1} = rac{6 - 2\sqrt{5}}{4} \), but we earlier derived it correctly as \( rac{3 - \sqrt{5}}{2} \), so it checks.
Therefore, the solutions are all real pairs where \( x = \sqrt{5} y \) or \( x = -\sqrt{5} y \), \( y
oxed{(x, y) ext{ such that } x = \pm \sqrt{5} y, \; y
Question:** A historian analyzing a manuscript finds an expression involving complex numbers \( z \) and \( w \) satisfying \[ rac{z + w}{z - w} + rac{z - w}{z + w} = 2. \] Determine the value of \( \left| rac{z}{w}
ight| \), assuming \( w

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