Difference between revisions of "2023 AMC 12B Problems/Problem 12"
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Thus, the magnitude of z is <math> \sqrt{a^{2}+b^{2}} = \sqrt{50} = 5\sqrt{2}</math> | Thus, the magnitude of z is <math> \sqrt{a^{2}+b^{2}} = \sqrt{50} = 5\sqrt{2}</math> | ||
<math>=\text{\boxed{\textbf{(E) }5\sqrt{2}}}</math> | <math>=\text{\boxed{\textbf{(E) }5\sqrt{2}}}</math> | ||
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Revision as of 17:00, 15 November 2023
Problem
For complex number and (where ), define the binary operation
Suppose is a complex number such that . What is ?
Solution 1
let = . . This is equal to Since the real values have to be equal to each other, Simple algebra shows , so is . The imaginary components must also equal each other, meaning , or . This means . Thus, the magnitude of z is
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