2004 AMC 12B Problem 16

Problem 16 of 25IntermediateAlgebra Geometry

A function

f

is defined by

f(z) = i\overline{z},

where

i = \sqrt{-1}

and

\overline{z}

is the complex conjugate of

z.

How many values of

z

satisfy both

|z| = 5

and

f(z) = z?

Show solution

Solution

Writing

z = x + iy,

we get

f(z) = i(x - iy) = y + ix.

Setting

f(z) = z

gives

y = x,

which is a line through the origin. The condition

|z| = 5

is a circle, and a line through the center meets the circle in

2

points. Thus, the correct answer is C.

Tagged: complex number · coordinate geometry