Bunuel
Line L is a line in the x-y plane. Does line L pass through the second quadrant?
(1) Line L has a positive slope
(2) At its closest approach, Line L is never closer than 3 units to the origin.
Kudos for a correct solution.Statement #1: Line L has a positive slope
Case one — line below origin goes through quadrants III, IV, and I.
Attachment:
gsdsq_img9.png [ 31.84 KiB | Viewed 6073 times ]
Case two — line through origin goes through just quadrants III & I.
Attachment:
gsdsq_img10.png [ 33.84 KiB | Viewed 6074 times ]
Case three — line below origin goes through quadrants III, II, and I.
Attachment:
gsdsq_img11.png [ 25.02 KiB | Viewed 6025 times ]
Some lines with positive slopes go through Quadrant II, and some do not. Statement #1, alone and by itself, is insufficient.
Statement #2: At its closest approach, Line L is never closer than 3 units to the origin.
In the diagram below, the red circle has a radius of three, and no line crosses within this circle — thus, every line in this diagram meets the condition given in Statement #2.
Attachment:
gsdsq_img12.png [ 70.65 KiB | Viewed 6009 times ]
All lines with negative slopes pass through Quadrant II. Vertical lines pass through Quadrant II only if they are to the left of the y-axis. Horizontal lines pass through Quadrant II only if they are above the x-axis. As we saw in statement #1, some lines with positive slope pass through Quadrant II and some don’t. Statement #2, alone and by itself, is insufficient.
Statements #1 & #2 combined: Now, we are confined to lines with positive slopes and we also have the red circle. The following diagram tells the whole story:
Attachment:
gsdsq_img13.png [ 23.95 KiB | Viewed 5983 times ]
Both of these lines meet both conditions — positive slopes and outside the red circle. One of them passes through Quadrant II, and one of the them doesn’t. Therefore, even if we know both conditions are true, that will not answer the prompt question. Combined, the statements are still insufficient.
Answer = E.