Re S99-20
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16 Sep 2014, 01:53
Official Solution:
For \(ab\) to be positive, \(a\) and \(b\) must have the same sign. Coordinates have the same sign in the first and third quadrants (upper right and lower left, following the standard numbering of quadrants). Thus, the question can be rephrased as "Do the lines intersect in either the first or third quadrant?"
We also know that the x-intercept of \(k\) and the y-intercept of \(m\) are both positive. This condition restricts the lines in the following way: \(k\) intersects the x-axis to the right of \((0, 0)\), and \(m\) intersects the y-axis above \((0, 0)\).
However, at this point, since the lines could slope in any direction, the intersection point could fall anywhere in the coordinate plane.
Statement (1): SUFFICIENT. A line with both a positive x-intercept and a negative y-intercept can only pass through the first, third, and fourth quadrants. Any line with a positive x-intercept must pass through the first and fourth quadrants (the quadrants on either side of the positive half of the x-axis). Likewise, any line with a negative y-intercept must past through the third and fourth quadrants (the quadrants on either side of the negative half of the y-axis). Finally, it is impossible for any line to pass through all four quadrants - to do so, the line would have to change slope. Thus, line \(k\) passes through quadrants I, III, and IV. Draw a coordinate plane and several versions of line \(k\); this result will become apparent.
Similarly, we can see that line \(m\), which has both a negative x-intercept and a positive y-intercept, must pass through the first, second, and third quadrants. Having a negative x-intercept means that the line passes through quadrants II and III (on either side of the negative half of the x-axis). Likewise, having a positive y-intercept means that the line passes through quadrants I and II (on either side of the positive half of the y-axis).
Finally, since the lines intersect, they must do so in a quadrant they both reach - that is, only quadrant I or quadrant III. Thus, we can answer "Yes" to our rephrased question.
Statement (2): SUFFICIENT. This condition, explicitly stating that both lines have positive slopes, turns out to lead to the first condition when we also consider the information given to us in the question stem. Consider line \(k\) alone. We know that its x-intercept is positive, meaning that the line goes through \((x, 0)\), where \(x\) is positive. Since the slope of the line is positive, we can conclude that the y-intercept is negative. The reason is this: if the coordinates of the y-intercept are \((0, y)\), then we can write the slope of the line as \(\frac{x - 0}{0 - y}\). We need this fraction to be positive, and the top is definitely positive; thus, the bottom must be positive as well, forcing \(y\) to be negative.
By a similar argument, if we know that line \(m\) has a positive y-intercept and a positive slope, then we can conclude that the line has a negative x-intercept.
Thus, the result is the same as for the first statement.
Answer: D