In the x-y coordinate plane, what is the minimum distance between a po

I'm simply guessing, yet again.

(I) Tells us that there's a distance of 4 between the intercepts at X=0. Not sufficient as the slope might not be equal (in which case the distance is constant).

(II) Tells us that the slope is either 2 or -2 for the both. If both are negative, then we could answer the question because the distance would be 4. If both are different, then they will intersect and then it would be 0.

As several possible options are available, the answer should be E.

I am slightly tempted to take the Option E. I know bunuel's questions are never easy.

My though process.

1. The absolute difference between the Y intercept of the lines is 4. The lines can be parallel or even the slant lines(having slope). Not sufficient.

2. The absolute value of the slopes of the lines is 2.

The absolute of a number has two forms + and -

Lets say One line is Y=2x+1 and another line Y=-2x+6; (Testing one positive and one negative; Lines interesect)


Lets say one line is Y=2x+1 and another line is Y= 2x+6; (Testing both positive; Parallel lines; Distance same.


Lets say one line is Y=-2x+1 and another line is Y=-2x+6 (Testing both slopes negative; Parallel lines;Distance same)


Both options together,

Y= 2x+3; Y= -2x-1 (interesecting lines)

Y=2x+0; Y=-2x-4

Y=2x+2 ; Y=2x-2 (parallel lines)


Hence My choice is E.

Statement (I) just means that at the y-axis, these two lines are 4 units apart, but tells us nothing about the slope. They could be perpendicular lines (min distance 0, where they intersect) or parallel lines (constant distance of 4). Not sufficient.

Statement (II) tells us that the slopes are either 2 or -2, though not necessarily which is which. Again, we could have parallel lines or intersecting lines. Not sufficient.

Statement (I) + (II) still do not tell us if the lines are parallel or intersecting, yielding possible answers of 4 or 0. Not sufficient.

E is the answer.

MANHATTAN GMAT OFFICIAL SOLUTION:

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 .

took me some time to double check and visualize...
1. distance between interceptors is 4 units. suppose we have 2 parallel horizontal lines. the distance is always 4. but if the lines intersect each other, then the minimum distance is 0.
2. slope is either positive 2, or negative 2. we can have 2 parallel lines, or we can have 2 intersecting lines. not sufficient.

1+2 we can have parallel lines or we can have two intersecting lines.

E

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