DS If line L in XY coordinate Plane has a positive slope…

[vak3e]

If line L in the xy-coordinate plane has a positive slope, what is the x-intercept of L ?

(1) There are different points (a, b) and (c, d) on line L such that ad = bc.

(2) There are constants m and n such that the points (m, n) and (-m, -n) are both on line L.


Hi! I couldn't figure out how to solve this in an elegant way. The algebraic approach for (1) in the MGMAT CAT explanations makes sense, but it would be too long to do on the exam. I ended up just picking a few numbers and sort of just 'reasoning' my way through this question, however, I was about 40-50% sure that I had actually gotten it wrong. It'd be great to see how an expert might approach such a problem.

Thank you!

[RonPurewal]

In a situation that's unusual, like this one"”or, more generally, in any situation in which you aren't quite sure what the next steps are"”just start investigating, however you can.
In the case of DS problems, this usually means "plug in a bunch of cases, and watch what they do."

With statement 1 here, after you plug in a handful of cases, it will become quite clear that you're always going to end up with points that are on the same straight line through the origin. (It will be even more apparent if you draw the points"”don't forget that you get graph paper on the test!)

[RonPurewal]

As far as non-"plugging" solutions"”
You may recognize "ad = bc" as the equation that results from cross-multiplying a proportion. I.e., if you start with a/b = c/d, you'll get ad = bc. Therefore, statement 1 can be rephrased as a/b = c/d.

When we do this, we have to be careful to make sure that neither b nor d is zero; if those were possible, we'd have to treat them as separate cases.
So, let's try to let b = 0. In that case, either a = 0 or d = 0 (to make ad = bc true). If a = 0, then you have the point (0, 0) on the line"”the same intercept you get in other cases. If d = 0, then you have (a, 0) and (c, 0) on the line"”a situation that's impossible, given that the line has a positive slope.
So, no need to worry about the zero cases: either they don't work in the first place, or they give the same result as other cases.

OK, back to a/b = c/d.
This means that the ratio of the x and y coordinates is the same for the two points. If that's true, then they are on the same line through (0, 0), because y is the same multiple of x both times. (It's a direct proportion.)