AMC12 2018 B

AMC12 2018 B · Q3

AMC12 2018 B · Q3. It mainly tests Linear equations, Coordinate geometry.

A line with slope 2 intersects a line with slope 6 at the point (40, 30). What is the distance between the x-intercepts of these two lines?

一条斜率为2的直线与一条斜率为6的直线相交于点(40, 30)。这两条直线的x轴截距之间的距离是多少？

(A) 5 5

(B) 10 10

(D) 25 25

(E) 50 50

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): The line with slope 2 containing the point $(40,30)$ has the equation $y-30=2(x-40)$. Similarly, the line with slope 6 containing the point $(40,30)$ has the equation $y-30=6(x-40)$. To find the $x$-intercepts of these two lines, let $y=0$ and solve for $x$ separately in each of these two equations. With the first equation the $x$-intercept is $25$, and with the second equation the $x$-intercept is $35$. Thus the distance between the two $x$-intercepts is $|25-35|=10$.

答案（B）：斜率为 $2$ 且经过点 $(40,30)$ 的直线方程为 $y-30=2(x-40)$。类似地，斜率为 $6$ 且经过点 $(40,30)$ 的直线方程为 $y-30=6(x-40)$。为求这两条直线的 $x$ 轴截距，令 $y=0$，并分别在这两个方程中解出 $x$。由第一个方程得 $x$ 截距为 $25$，由第二个方程得 $x$ 截距为 $35$。因此两个 $x$ 截距之间的距离为 $|25-35|=10$。

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