2021 AMC 10A Spring 第 24 题

Question

一个四边形的内部由图像

(x+ay)^2=4a^2

和

(ax-y)^2=a^2

围成，其中

a

是正实数。对所有

a > 0

，这个区域的面积用

a

表示是多少？

The interior of a quadrilateral is bounded by the graphs of

(x+ay)^2=4a^2

and

(ax-y)^2=a^2,

where

a

is a positive real number. What is the area of this region in terms of

a,

valid for all

a > 0?

\dfrac{8a^2}{(a+1)^2}

\dfrac{4a}{a+1}

\dfrac{8a}{a+1}

\dfrac{8a^2}{a^2+1}

\dfrac{8a}{a^2+1}

答案：D

知识点：平行线距离公式矩形

难度评级：1720

小提示：

每个平方方程都表示一对平行直线。

Each squared equation represents a pair of parallel lines

大提示：

两组平行线互相垂直，所以面积是两组平行线距离的乘积。

The two pairs of lines are perpendicular, so the area is the product of the two distances between parallel lines

解答：

注意，每个方程都会给出两条平行直线。

(x + ay)^2 = 4a^2

给出两条直线

x + ay - 2a = 0

和

x + ay + 2a = 0\text{。}

这两条直线的斜率都是

-\dfrac{1}{a}

。

类似地，

(ax-y)^2 = a^2

给出直线

ax - y - a = 0

和

ax - y + a = 0\text{。}

这些直线的斜率为

a

。

两组直线互相垂直，因此围成一个矩形。

回忆两条平行线

\begin{cases} Ax+By+C_1=0 \\ Ax+By+C_2=0 \end{cases}

之间的距离

d

为

d = \dfrac{\mid C_2 - C_1 \mid}{\sqrt{A^2 + B^2}}\text{。}

用这个公式，第一组平行线之间的距离为

\dfrac{4a}{\sqrt{a^2 + 1}}\text{。}

类似地，第二组平行线之间的距离为

\dfrac{2a}{\sqrt{a^2 + 1}}\text{。}

这两个距离就是矩形的边长，相乘得面积

\dfrac{8a^2}{a^2 + 1}\text{。}

所以正确答案是 D。

Note that each of the equations yields two parallel lines.

(x + ay)^2 = 4a^2

results in the two lines

x + ay - 2a = 0

and

x + ay + 2a = 0.

Both of these lines have a slope of

-\dfrac{1}{a}.

Similarly,

(ax-y)^2 = a^2

results in the lines

ax - y - a = 0

and

ax - y + a = 0.

These lines have slope

a.

Note that each pair of lines is perpendicular to the other pair of lines. This shows that the equations form a rectangle.

Recall that the formula for the distance

d

between two parallel lines

\begin{cases} Ax+By+C_1=0 \\ Ax+By+C_2=0 \end{cases}

is

d = \dfrac{\mid C_2 - C_1 \mid}{\sqrt{A^2 + B^2}}.

Using this formula, we get that the distance between the first pair of lines is

\dfrac{4a}{\sqrt{a^2 + 1}}.

Similarly, the distance between the second pair of lines is

\dfrac{2a}{\sqrt{a^2 + 1}}.

These are the side lengths of the rectangle. Multiplying yields the area

\dfrac{8a^2}{a^2 + 1}.

Thus, D is the correct answer.

2021 AMC 10A Spring 第 24 题

答案：D

难度评级：1720

小提示：

大提示：

解答：

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