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

2004 AMC 12B Problem 16 — Solution

2004 AMC 12B Problem 16

Below is the professionally curated solution for Problem 16 of the 2004 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12B solutions, or check the answer key.

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Concepts:complex number coordinate geometry

Difficulty rating: 1610

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 $f(z) = z?$

$0$

$1$

$2$

$4$

$8$

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.

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