Difference between revisions of "2021 AMC 12B Problems/Problem 18"
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| − | The answer being in the form <math>z+\frac 6z</math> means that there are two solutions, some complex number and its complex conjugate. <cmath>a+bi = \frac{6}{a-bi}</cmath> <cmath>a^2+b^2=6</cmath> We should then be able to test out some ordered pairs of (a, b). After testing it out, we get the ordered pairs of <math>(-1, \sqrt{5})</math> and its conjugate <math>(-1, -\sqrt{5})</math>. Plugging this into answer format gives us <math>\boxed{\textbf{(A) }-2}</math> ~Lopkiloinm | + | The answer being in the form <math>z+\frac 6z</math> means that there are two solutions, some complex number and its complex conjugate. <cmath>a+bi = \frac{6}{a-bi}</cmath> <cmath>a^2+b^2=6</cmath> We should then be able to test out some ordered pairs of <math>(a, b)</math>. After testing it out, we get the ordered pairs of <math>(-1, \sqrt{5})</math> and its conjugate <math>(-1, -\sqrt{5})</math>. Plugging this into answer format gives us <math>\boxed{\textbf{(A) }-2}</math> ~Lopkiloinm |
Revision as of 19:57, 11 February 2021
Problem 18
Let 
be a complex number satisfying 
What is the value of 
Solution
Solution 1
The answer being in the form 
means that there are two solutions, some complex number and its complex conjugate. ![\[a+bi = \frac{6}{a-bi}\]](http://latex.artofproblemsolving.com/5/f/3/5f333ff36bfce8cc1cbef23b56cdc66c678671eb.png)
![\[a^2+b^2=6\]](http://latex.artofproblemsolving.com/a/0/0/a006fc3bc9d7d0863ef53f8dd41755cca1848ac8.png)
We should then be able to test out some ordered pairs of 
. After testing it out, we get the ordered pairs of 
and its conjugate 
. Plugging this into answer format gives us 
~Lopkiloinm
