Difference between revisions of "2023 AMC 12B Problems/Problem 12"
Failure.net (talk | contribs) (→Solution 1) |
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This is equal to <math>z^{2} + 40 = a^{2}-b^{2}+40+2abi</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> | + | 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>. | Simple algebra shows <math>b^{2} = 40</math>, so <math>b</math> is <math>2\sqrt{10}</math>. | ||
Revision as of 17:01, 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
~Failure.net