1997 AMC 12 第 12 题

Question

若

m

和

b

是实数，且

mb\gt0

，则方程为

y=mx+b

的直线不可能经过点

If

m

and

b

are real numbers and

mb\gt0,

then the line whose equation is

y=mx+b

cannot contain the point

(0,1997)

(0,-1997)

(19,97)

(19,-97)

(1997,0)

条件

mb\gt0

表示斜率与纵截距同号

The condition

mb\gt0

means the slope and vertical intercept have the same sign

逐一代入各点；正的

x

轴截距会迫使斜率与截距异号

Substitute each point; a positive

x

-intercept forces the slope and intercept to have opposite signs

若

(1997,0)

在直线上，则

0=1997m+b

，所以

b=-1997m

，从而

mb=-1997m^2\lt0

，产生矛盾。对于其余每个点，都可以选取适当的同号

m,b

使直线经过该点。因此正确答案是 E。

If

(1997,0)

were on the line, then

0=1997m+b,

so

b=-1997m

and

mb=-1997m^2\lt0,

a contradiction. Each other point can occur for suitable same-sign

m,b.

Thus the correct answer is E.

其他年份的第 12 题