Manhattan GMAT Challenge Problem of the Week – 2 July 2012
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Question
In the x-y coordinate plane, what is the minimum distance between a point on line L and a point on line M?
(1) The absolute value of the difference between the y-intercepts of the two lines is 4.
(2) The absolute values of the slopes of the two lines are both equal to 2.A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.
Answer
Imagine two lines in a plane. Either they cross, or they don’t. If they cross, then the “minimum distance” between a point on one line and a point on the other is zero (because you can pick the same point for both lines, namely the intersection point).
If the lines don’t cross, then they’re parallel to each other, and the minimum distance between a point on one line and a point on the other line is what you normally think of as the distance between two parallel lines – go “straight across the street” from one line to the other.
Statement 1: NOT SUFFICIENT. One line goes through the y-axis at a point 4 units away from where the second line goes through the y-axis. However, you have no idea whether these lines cross each other somewhere else, so there’s no way to know the minimum distance between a point on one line and a point on the other.
Statement 2: NOT SUFFICIENT. The lines could both have slope 2 (and therefore be parallel), they could both have slope -2 (and also be parallel), or one could have slope 2 and the other could have slope -2 (in which case they would cross). We don’t know whether the lines cross or not, so again, we can’t know the minimum distance between the points.
Statements 1 & 2 together: STILL NOT SUFFICIENT. Even together, the statements don’t narrow down the cases sufficiently. If one line has slope -2 and y-intercept of 4, while the other line has slope 2 and y-intercept of 0, then the lines will cross and the minimum distance between them is zero. But if the two lines both have slope -2 (and the respective intercepts), then the minimum distance between them is not zero.
The correct answer is E .
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