2009 AMC 12B Problems/Problem 23
A region in the complex plane is defined by A complex number is chosen uniformly at random from . What is the probability that is also in ?
First, turn into polar form as . Restated using geometric probabilities, we are trying to find the portion of a square enlarged by a factor of and rotated degrees that lies within the original square. This skips all the absolute values required before. Finish with the symmetry method stated above.
We multiply and to get Since we want to find the probability that this number is in , we need the real and complex coefficients of this number to be less than or equal to or greater than or equal to This gives us the equations and Now, we see that we can solve this by graphing. We can graph our barriers and to form a by square centered at the origin. Graphing our two equations gives us the four lines The square that is formed is the region that satisfies these four equations. Now, the barriers and this square gives us an octagon as the desired region. The area of this octagon is the total area of the square minus the 4 small triangles on each corner, each with area. Therefore, the octagon has area of Finally, to find the probability of it working, we find the area of the octagon divided by the area of the entire square which is or
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