Question

函数 $f$ 定义为 $f(z) = i\overline{z}$ ，其中 $i = \sqrt{-1}$ ， $\overline{z}$ 是 $z$ 的复共轭。有多少个 $z$ 同时满足 $|z| = 5$ 和 $f(z) = z$ ？

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 $f(z) = z?$

$0$

$1$

$2$

$4$

$8$

答案：C

知识点：复数坐标几何

难度评级：1610

解答：

写 $z = x + iy$ 则 $f(z) = i(x - iy) = y + ix$ 。令 $f(z) = z$ 得 $y = x$ ，这是一条过原点的直线。条件 $|z| = 5$ 是一个圆，过圆心的直线与圆相交于 $2$ 个点。

因此正确答案是 C。

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.

2004 AMC 12B 第 16 题

答案：C

难度评级：1610

解答：

其他年份的第 16 题