AMBIGUITY

[Nov1907]

In the xy-coordinate system, what is the slope of the line that goes through the origin and is equidistant from the two points P = (1, 11) and Q = (7, 7)?

2
2.25
2.50
2.75
3
(Source: MGMAT Cat - The incredible line)

I misunderstood this question to mean that the perpendicular distance from the points to the line are equal. Is there a reason that this is wrong? Could an instructor please review this and get back to me. Please!

[TheChakra]

In the xy-coordinate system, what is the slope of the line that goes through the origin and is equidistant from the two points P = (1, 11) and Q = (7, 7)?

2
2.25
2.50
2.75
3
(Source: MGMAT Cat - The incredible line)

I misunderstood this question to mean that the perpendicular distance from the points to the line are equal. Is there a reason that this is wrong? Could an instructor please review this and get back to me. Please!

Your interpretation is not wrong. The line is equidistant (meaning perpendicular distance from the line). PQ is also bisected by the line from origin. Actually, there are only 2 cases

1. PQ is bisected
2. PQ is parallel to the line in question

The proof given in MCAT isn't really convincing. How do you differentiate between the two cases? I am not sure. You could have solved for both the cases and found that no answer for case 2 is present.

You can probably say that since PQ are both in Q1 and since PQ has a -ve slope, a line cannot possibly be parallel to PQ and still go through origin. Hence 1 is the only scenario left.

[StaceyKoprince]

Yep - sketch the graph and realize that, in one option you have a positive slope and in the other you have a negative slope. Glance at the choices and realize that only positive slopes are given, so only solve for that case (which is the "perpendicular case").

It is not unusual at all to see that type of set up on the official test - there are multiple scenarios but only one of the possible answers is given in the choices.

[ETA: Our curriculum director, Chris, pointed out to me that, given the wording of the problem, the negative-slope line does not qualify as a possibility. The problem says the line is "equidistant" from the two points. Any point on the positive-slope line is equidistant from the two points. That's not true for the negative-slope line. So the only one that qualifies is the positive-slope line. What I wrote above is still true in general for the test but does not need to be applied to this particular problem (though, if you didn't notice the implications of the word "equidistant" then that's another way to get yourself to the realization that you need to do the positive-slope line).]