1988 AIME Problems/Problem 11
Let be complex numbers. A line in the complex plane is called a mean line for the points if contains points (complex numbers) such that For the numbers , , , , and , there is a unique mean line with -intercept 3. Find the slope of this mean line.
Each lies on the complex line , so we can rewrite this as
Matching the real parts and the imaginary parts, we get that and . Simplifying the second summation, we find that , and substituting, the answer is .
We know that
And because the sum of the 5 's must cancel this out,
We write the numbers in the form and we know that
The line is of equation . Substituting in the polar coordinates, we have .
Summing all 5 of the equations given for each , we get
Solving for , the slope, we get
The mean line for must pass through the mean (the center of mass) of these points, which, if we graph these points on the complex plane, is . Since we now have two points, namely that one and , we can simply find the slope between them, which is by the good ol' slope formula.
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