1999 AIME Problem 9

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

Small hint:

The condition says

|f(z) - z| = |f(z)|

for every

z;

write both sides as a modulus times

|z|

Big hint:

You need

|a - 1 + bi| = |a + bi|,

which pins down

a;

then use

a^2 + b^2 = 64

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.