Question

A function $f$ is defined on the complex numbers by $f(z) = (a + bi)z,$ where $a$ and $b$ 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 $|a + bi| = 8$ and that $b^2 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$

Answer

Answer: 259

Concepts:complex number algebraic manipulation

Difficulty rating: 2270

Solution:

The condition is $|f(z) - z| = |f(z)|$ for all $z,$ that is, $|(a - 1 + bi)z| = |(a + bi)z|.$ Dividing by $|z|$ (for $z \ne 0$ ) gives $|a - 1 + bi| = |a + bi|,$ so $(a - 1)^2 + b^2 = a^2 + b^2,$ which forces $a = \frac{1}{2}.$

Since $|a + bi| = 8,$ we have $a^2 + b^2 = 64,$ so $b^2 = 64 - \frac{1}{4} = \frac{255}{4}.$ As $\gcd(255, 4) = 1,$ the answer is $255 + 4 = 259.$

1999 AIME Problem 9

Answer: 259

Difficulty rating: 2270

Solution:

Problem 9 in Other Years