2018 AMC 12A 第 22 题

Question

方程

z^2 = 4 + 4\sqrt{15}\,i

和

z^2 = 2 + 2\sqrt{3}\,i

的解（其中

i = \sqrt{-1}

）在复平面中构成一个平行四边形的顶点。这个平行四边形的面积可写成

p\sqrt{q} - r\sqrt{s}

，其中

p

、

q

、

r

和

s

是正整数，且

q

和

s

都不被任何质数的平方整除。

p + q + r + s

是多少？

The solutions to the equations

z^2 = 4 + 4\sqrt{15}\,i

and

z^2 = 2 + 2\sqrt{3}\,i,

where

i = \sqrt{-1},

form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form

p\sqrt{q} - r\sqrt{s},

where

p,

q,

r,

and

s

are positive integers and neither

q

nor

s

is divisible by the square of any prime number. What is

p + q + r + s?

20

21

22

23

24

答案：A

知识点：复数鞋带公式

难度评级：2270

小提示：

写

z = a + bi

，通过比较实部和虚部解

z^2 = 4 + 4\sqrt{15}\,i

；另一个方程也同样处理

Solve

z^2 = 4 + 4\sqrt{15}\,i

by writing

z = a + bi

and matching real and imaginary parts; do the same for the other equation

大提示：

四个顶点是

\pm z_1

和

\pm z_2

；对它们的坐标使用鞋带公式

The four vertices are

\pm z_1

and

\pm z_2;

apply the shoelace formula to their coordinates

解答：

写

z = a + bi

，由

(a + bi)^2 = 4 + 4\sqrt{15}\,i

得

a^2 - b^2 = 4

和

2ab = 4\sqrt{15}

。因此

a^4 - 4a^2 - 60 = 0

，即

(a^2 - 10)(a^2 + 6) = 0

，得

a = \pm\sqrt{10}

，

b = \pm\sqrt{6}

。第一个方程给出的顶点为

\pm(\sqrt{10} + \sqrt{6}\,i)

。对

z^2 = 2 + 2\sqrt{3}\,i

用同样方法得到

\pm(\sqrt{3} + i)

。

对

(\sqrt{10}, \sqrt{6})

，

(\sqrt{3}, 1)

，

(-\sqrt{10}, -\sqrt{6})

，

(-\sqrt{3}, -1)

使用鞋带公式，面积为

6\sqrt{2} - 2\sqrt{10}

。因此

p + q + r + s

= 6 + 2 + 2 + 10

= 20

。

所以正确答案是 A。

Writing

z = a + bi

with

(a + bi)^2 = 4 + 4\sqrt{15}\,i

gives

a^2 - b^2 = 4

and

2ab = 4\sqrt{15}.

Then

a^4 - 4a^2 - 60 = 0,

so

(a^2 - 10)(a^2 + 6) = 0,

yielding

a = \pm\sqrt{10},

b = \pm\sqrt{6}.

The vertices from the first equation are

\pm(\sqrt{10} + \sqrt{6}\,i).

The same method on

z^2 = 2 + 2\sqrt{3}\,i

gives

\pm(\sqrt{3} + i).

Applying the shoelace formula to

(\sqrt{10}, \sqrt{6}),

(\sqrt{3}, 1),

(-\sqrt{10}, -\sqrt{6}),

(-\sqrt{3}, -1)

gives area

6\sqrt{2} - 2\sqrt{10}.

Thus

p + q + r + s

= 6 + 2 + 2 + 10

= 20.

Thus, the correct answer is A.

2018 AMC 12A 第 22 题

答案：A

难度评级：2270

小提示：

大提示：

解答：

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