Difference between revisions of "2023 AMC 12B Problems/Problem 12"
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<math>\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }\sqrt{5}\qquad\textbf{(D) }\sqrt{10}\qquad\textbf{(E) }5\sqrt{2}</math> | <math>\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }\sqrt{5}\qquad\textbf{(D) }\sqrt{10}\qquad\textbf{(E) }5\sqrt{2}</math> | ||
+ | |||
+ | ==Solution 1== | ||
+ | let <math>z</math> = <math>a+bi</math>. | ||
+ | <math>z \odot z a^{2}+b^{2}i</math>. | ||
+ | This is equal to <math>z^{2} + 40 = a^{2}-b^{2}+40+2abi</math> | ||
+ | Since the real values have to be equal to each other, <math>a^{2}-b^{2}+40 = a^{2}</math> | ||
+ | Simple algebra shows <math>b^{2} = 40</math>, so <math>b</math> is <math>2\sqrt{10}</math>. | ||
+ | The imaginary components must also equal each other, meaning <math>b^{2} = 2ab</math>, or <math>b = 2a</math>. This means <math>a = \frac{b}{2} = \sqrt{10}</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> | ||
+ | |||
+ | ~zhenghua | ||
+ | |||
+ | |||
+ | ~Failure.net |
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
~zhenghua
~Failure.net