Difference between revisions of "2022 AMC 12A Problems/Problem 13"
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If <math>z</math> is a complex number and <math>z = a + bi</math>, then the magnitude (length) of <math>z</math> is <math>\sqrt{a^2 + b^2}</math>. Therefore, <math>z_1</math> has a magnitude of 5. If <math>z_2</math> has a magnitude of at most one, that means for each point on the segment given by <math>z_1</math>, the bounds of the region <math>\mathcal{R}</math> could be at most 1 away. Alone the line, excluding the endpoints, a rectangle with a width of 2 and a length of 5, the magnitude, would be formed. At the endpoints, two semicircles will be formed with a radius of 1 for a total area of <math>\pi \approx 3</math>. | If <math>z</math> is a complex number and <math>z = a + bi</math>, then the magnitude (length) of <math>z</math> is <math>\sqrt{a^2 + b^2}</math>. Therefore, <math>z_1</math> has a magnitude of 5. If <math>z_2</math> has a magnitude of at most one, that means for each point on the segment given by <math>z_1</math>, the bounds of the region <math>\mathcal{R}</math> could be at most 1 away. Alone the line, excluding the endpoints, a rectangle with a width of 2 and a length of 5, the magnitude, would be formed. At the endpoints, two semicircles will be formed with a radius of 1 for a total area of <math>\pi \approx 3</math>. | ||
Therefore, the total area is <math>5(2) + \pi \approx 10 + 3 = 13</math> (A) | Therefore, the total area is <math>5(2) + \pi \approx 10 + 3 = 13</math> (A) | ||
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Revision as of 15:41, 12 November 2022
Problem
Let 
be the region in the complex plane consisting of all complex numbers 
that can be written as the sum of complex numbers 
and 
, where lies on the segment with endpoints 
and 
, and has magnitude at most 
. What integer is closest to the area of ?
Solution
If is a complex number and 
, then the magnitude (length) of is 
. Therefore, has a magnitude of 5. If has a magnitude of at most one, that means for each point on the segment given by , the bounds of the region could be at most 1 away. Alone the line, excluding the endpoints, a rectangle with a width of 2 and a length of 5, the magnitude, would be formed. At the endpoints, two semicircles will be formed with a radius of 1 for a total area of 
.
Therefore, the total area is 
(A)
-juicefruit
