In the xy-plane, the line k passes through the origin

[Jenna]

In the xy-plane, the line k passes through the origin and through the point (a,b), where ab does not equal 0. Is b positive?

1) The slope of line k is negative

2) a<b

I answered A because I thought that since we know that the line passes through (0,0) and has a negative slope, b has to be negative. The answer to the question is "no", and A is sufficient. However, the official answer is C. Can you please help explain why?

[Venu]

Since the line passes through origin and (a,b) then the equation of line is

ay=bx
or the slope is (b/a)

Option a) slope is negative , then (b/a)<0 .This would lead to 2 cases :-

i) b < 0 when a is positive
ii) b > 0 when a is negative.

So not sufficient

Option b) a<b Does not tells me anything

however by combining both of them, rules out the case b < 0 when a is positive (deduced from the option a)

So we are left with b>0 when a is negative.

Please comment if my approach is correct.

[StaceyKoprince]

Nice, Venu.

Any time you have a geometry problem, sketch what's going on. So, sketch a coordinate plane right now (it's okay, I'll wait ) It's a lot harder for them to trick us when we can visualize what's going on.

Put a dot at the origin. To have a negative slope (per statement 1), you draw the line from the upper left somewhere to the lower right. This places point (a,b) either in the 2nd or the 4th quadrant. In the 2nd quadrant, b is positive. In the 4th quadrant, b is negative. b could be either pos or neg - can't tell.

Statement 2, by itself, tells us nothing definitive - b could be pos or neg.

Once I put them together and see that a<b, now I know (a,b) has to be in quadrant 2, because in quad 4, x is always pos and y is always neg. So C is the right answer.

[Guest]

Thank you, Stacey, for the excellent explanation! The visualization technique you mentioned certainly helps here!

[GMAT 5/18]

Yes, for questions involving the coordinate plane, I find it is almost essential to draw it out. Even some of those difficult perpendicular bi-sector questions become relatively easy when you draw out the question (I am referring to some of those questions in the OG 11th ed.).

Thanks Venu and Stacey!

[vinversa]

Chiming into Venu's version


In the xy-plane, the line k passes through the origin and through the point (a,b), where ab <> 0. Is b positive?

(1) The slope of line k is negative
(2) a<b


From question we get:
Line k = {y-y1 = m(x - x1)} passes through origin (0,0)
So line k = y-0 = m(x-0)
Line K is y=mx

Line k passes through point (a,b).
Therefore, substitute (a,b) in y=mx
We get b=ma


(1) The slope of line k is negative
b=-ma
Not sufficient


(2) a<b
b=ma
a=2 b=3
a=-3 b=-2
Not Sufficient


(1) & (2)
b=-ma
a<b

let us say |m|=1
b=-1.a
if a=2 then b=-2 (not possible since a<b)
if a=-2 then b = 2 (possible since a<b)

a has to be negative to satisfy a<b

Therefore b - turns out positive all the time.

HENCE [C] is the answer

[RonPurewal]

looks like both of the solutions on here are valid, but all of you reading this topic should make an effort to cultivate the sort of reasoning found in the first post (the one from "venu").

that sort of reasoning -- in which the student circumvents the need for complicated algebraic reasoning by thinking about the impact of number properties (in this case, the signs of the numbers in each quadrant) -- is usually much quicker and less messy than algebra, on those problems where it actually works.