2. 外语
  3. In the xy-plane, line A; is a line that does not pass through the origin. Which of the following statements individually provide

In the xy-plane, line A; is a line that does not pass through the origin. Which of the following statements individually provide

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问题 In the xy-plane, line A; is a line that does not pass through the origin.
Which of the following statements individually provide(s)sufficient additional information to conclude that the slope of line k is negative?
Indicate all such statements.
选项 A、The x-intercept of line k is twice the y-intercept of line k.
B、The product of the x-intercept and the y-intercept of line k is positive.
C、Line k passes through the points(a,b)and(r,s), where(a - r)(b - s)< 0.
答案A，B，C
解析 For questions involving a coordinate system, it is often helpful to draw a figure to visualize the problem situation. If you draw some lines with negative slopes in the xy-plane, such as those in the figure below, you see that for each line that does not pass through the origin, the x- and y-intercepts are either both positive or both negative. Conversely, you can see that if the x- and y-intercepts of a line have the same sign then the slope of the line is negative.

    You can use this fact to examine the information given in the first two statements. Remember that you need to evaluate each statement by itself.
    Choice A states that the x-intercept is twice the y-intercept, so you can conclude that both intercepts have the same sign, and thus the slope of line k is negative. So the information in Choice A is sufficient to determine that the slope of line k is negative.
    Choice B states that the product of the x-intercept and the y-intercept is positive. You know that the product of two numbers is positive if both factors have the same sign. So this information is also sufficient to determine that the slope of line k is negative.
    To evaluate Choice C, it is helpful to recall the definition of the slope of a line passing through two given points. You may remember it as "rise over run." If the two points are(a, b)and(r,s), then the slope is.
    Choice C states that the product of the quantities(a - r)and(b - s)is negative. Note that these are the denominator and the numerator, respectively, of , the slope of line k. So you can conclude that(a - r)and(b - s)have opposite signs and the slope of line k is negative. The information in Choice C is sufficient to determine that the slope of line k is negative.
    So each of the three statements individually provides sufficient information to conclude that the slope of line k is negative. The correct answer consists of Choices A, B, and C.
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