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

$u \otimes v = ac + bdi$

Suppose $z$ is a complex number such that $z\otimes z = z^{2}+40$ . What is $|z|$ ?

$\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }\sqrt{5}\qquad\textbf{(D) }\sqrt{10}\qquad\textbf{(E) }5\sqrt{2}$

let $z$ = $a+bi$ .

$z \otimes z = a^{2}+b^{2}i$ .

This is equal to $z^{2} + 40 = a^{2}-b^{2}+40+2abi$

Since the real values have to be equal to each other, $a^{2}-b^{2}+40 = a^{2}$ . Simple algebra shows $b^{2} = 40$ , so $b = \pm 2\sqrt{10}$ .

The imaginary components must also equal each other, meaning $b^{2} = 2ab$ , or $b = 2a$ . This means $a = \frac{b}{2} = \pm \sqrt{10}$ .

Thus, the magnitude of $z$ is $\sqrt{a^{2}+b^{2}} = \sqrt{10+40} = \sqrt{50} = 5\sqrt{2}$ $=\text{\boxed{\textbf{(E) }5\sqrt{2}}}$

~Failure.net

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

2023 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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All AMC 12 Problems and Solutions