A function f is defined by f(z) = iz, where i = √-1 and z is the complex conjugate of z. How many values of z satisfy both |z| = 5 and f(z) = z?

2004 AMC 12B Problem 16

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16.

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?

0

1

2

4

8

Write

z = x + iy

and compute

i\overline{z}

f(z) = z

forces

y = x,

a line; intersect it with the circle

|z| = 5

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.