1983 AHSME Problems/Problem 17
Problem
The diagram above shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of . Which one?
Solution
Write as , where we see from the diagram that and (as is outside the unit circle). We have , so, since , the reciprocal of has a positive real part and negative imaginary part. Also, the reciprocal has magnitude equal to the reciprocal of 's magnitude (since ); as 's magnitude is greater than , its reciprocal's magnitude will thus be between and , so its reciprocal will be inside the unit circle. Therefore, the only point shown which could be the reciprocal of is point .
See Also
1983 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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