1989 AMC 12 第 6 题

Question

若

a

、

b\gt0

，且第一象限中由坐标轴和直线

ax+by=6

围成的三角形面积为

6

，则

ab=

If

a,

b\gt0

and the triangle in the first quadrant bounded by the coordinate axes and the graph of

ax+by=6

has area

6,

then

ab=

3

6

12

108

432

求直线在两条坐标轴上的截距

Find the intercepts of the line on the two axes

将两个截距分别作为三角形的底和高

Use those intercepts as the base and height of the triangle

两个截距分别为

\frac{6}{a}

和

\frac{6}{b}

。因此三角形面积为

\frac12\cdot\frac6a\cdot\frac6b=\frac{18}{ab}=6\text{，}

所以

ab=3

。

所以正确答案是 A。

The intercepts are

\frac{6}{a}

and

\frac{6}{b}.

Therefore the triangle’s area is

\frac12\cdot\frac6a\cdot\frac6b=\frac{18}{ab}=6,

which gives

ab=3.

Thus the correct answer is A.

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