2023 AMC 12B Problems/Problem 12

Problem

For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$ ), define the binary operation

$u \earth v = ac \sum bdi$

$\textbf{(A) }\frac{9}{25}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{5}\qquad\textbf{(D) }\frac{25}{169}\qquad\textbf{(E) }\frac{4}{25}$

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2023 AMC 12B Problems/Problem 12

Problem