1999 AIME Problems/Problem 9
Problem
A function is defined on the complex numbers by
where
and
are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that
and that
where
and
are relatively prime positive integers. Find
Solution
Suppose we pick an arbitrary point on the line on the complex plane, say
. According to the definition of
, the image must be equidistant to
and
. These points lie on the line with slope
and which passes through
, so its graph is
. Substituting
and
, we get
.
By the Pythagorean Theorem, we have $\left(\frac{1}{2}\right_^2 + b^2 = 8^2 \Longrightarrow b^2 = \frac{255}{4}$ (Error compiling LaTeX. Unknown error_msg), and the answer is .
See also
1999 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |