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

Question

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 GmatPrep 9 - DS.JPG

Bunuel · Accepted Answer

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

(1) The slope of line k is negative. If slope is negative and the line passes through the origin, point (a,b) can be either in the II quadrant or in the IV (a and b have opposite signs). So, b can be positive or negative. Not sufficient.

(2) a < b. Not sufficient by itself.

(1)+(2) a < b and they have opposite signs, means b is positive (point lies in the second quadrant). Sufficient.

Answer: C.

Hope it helps.

BrentGMATPrepNow · Answer

Target question :    Is b (the y-coordinate of the point on the line) positive?

Given : Line k passes through the origin and through the point (a,b)

Statement 1 : The slope of line k is negative   
There are several lines and points that satisfy statement 1. Here are two:

Case a :
 https://i.imgur.com/w6TUtnj.png 
In this case, b (y-coordinate) is positive. So, the answer to the target question is  YES, b is positive

Case b :
 https://i.imgur.com/z5V6ZdN.png 
In this case, b (y-coordinate) is negative. So, the answer to the target question is  NO, b is NOT positive

Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : a < b 
There are several lines and points that satisfy statement 2. Here are two:

Case a :
 https://i.imgur.com/uHm11r8.png 
In this case, b (y-coordinate) is positive. So, the answer to the target question is  YES, b is positive

Case b :
 https://i.imgur.com/J247hxP.png 
In this case, b (y-coordinate) is negative. So, the answer to the target question is  NO, b is NOT positive

Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Statement 1 tells us that the slope of line k is negative. This means line k passes through quadrants II and IV. 
 https://i.imgur.com/I64iKvJ.png 
In quadrant II, a (the x-coordinate) is always negative, and b (the y-coordinate) is always positive
In quadrant IV, a (the x-coordinate) is always positive, and b (the y-coordinate) is always negative

Statement 2 tells us that a < b
This means that the point (a,b) must be in quadrant II  (because, all points in quadrant IV are such that the x-coordinate (a) is greater than the y-coordinate (b)  
If point (a,b) is in quadrant II, we can be certain that   b (the y-coordinate) is positive 
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

ezhilkumarank · Answer

If the line passes through (0,0) and (a,b), then the equation of the line is

(y-0)/(x-0) = (0-b)/(0-a)

(y/x) = (b/a). Hence y = (b/a)x.

Identified slope is (b/a).

Statement a says that the slope is negative. Thus m = -(b/a). However we cannot deduce whether a is -ve or b is -ve.

Statement 2: Says a<b but does not tell us the sign of a or b.

Combining both statements we know that a is less than b and than slope is negative.

Hence -(b/a) can be possible because of m = (b/(-a)) and that b is positive.

Answer is C. Combining both statements helps us answer the question.

vsaketram · Answer

Hi Bunuel,

If the line is passing through the origin, the equation can be either y=x or y=-x. Hence from option B, if a<b, it means that the equation of the line is y=-x and since a<b, a is negative and hence B is positive.

Please let me know the flaw in this thought flow.

Bunuel · Answer

Every line of the form y = mx passes through the origin, not only y = x and y = -x.

LogicGuru1 · Answer

Statement 1) The slope of line k is negative 
Using this and the stimulus we know that we have a line with negative slope that passes through the origin.
But the point (a,b) can lie either on Q II or on Q IV 
In Q II b is positive, But in Q IV b is negative.. so INSUFFICIENT

Statement 2) a<b
Now point "a" is less than point "b" - Apart from Q IV in which "point a" will always positive and "point" b is always negative, this condition "a<b" can happen in any three of Q I, Q II, Q III 
After all all it is saying is magnitude of a is less than magnitude of b .. Possible in Q I, Q II and Q III

Merging the two statements 
b has to be greater and line has to pass through Q II or IV, there is only one possible quadrant Q II where a is always -ve b is aways +ve. therefore b>a (always)
This satisfies both ur condition and thus we can say that YES B is positive.
Hence answer is C

rulingbear · Answer

is b positive?
 
1. b/a = negative, not sufficient, either b or a could be negative
2. a<b, not sufficient to determine sign of either  a or b

Combining 1 and 2 is enough to know b must be positive and a is negative. Hence C

SpyrosGkikopoulos · Answer

I solved it a little bit differently. i dont know whether it is correct but took me less time than the one presented above.
Stm 1: slope is negative ===> (b-0)/(a-0)<0 ===> either a>0 and b<0 or a<0 and b>0. Insf.
Stm 2: a<b insf

Both a<b ===> b>0 Ans C.

SchruteDwight · Answer

Statement 1)  \frac{b}{a}=m  as the  y -intercept is 0. Hence  \frac{b}{a}<0  but we don't know whether  a  or  b  is smaller than zero. Therefore, insufficient

Statement 2)  a<b  insufficient.

Statement 1 & 2)  a<0 \implies b>0

GloryBoy92 · Answer

Hi Guys,

What does "where ab ≠ 0" mean?

Kind Regards,

jkrishh7 · Answer

**************************************************************
Line k passes through the origin (0,0) and (a,b). Is b>0?

S1
The slope is negative ie (b-0)/(a-0) is less than 0
or
(b/a)<0 which implies b<0 or a<0... NOT SUFFICIENT

S2
a<b clearly does not give us any info about value or sign of a or b... NOT SUFFICIENT

Combine both statements
b<0 or a<0 (both cannot be) and a<b

If a<b then a is negative cleary and b is positive

Answer is C

CrackverbalGMAT · Answer

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

The general equation of the line that passes through origin is y = mx. where m is the slope.

Statement 1: The slope of line k is negative.

Since the slope is negative and the line K passes through origin, we can say that Line K passes through the Quadrant 2 and 4 only.
If point (a,b) in Quadrant 2, then a <0 and b >0 and if point (a,b) in Quadrant 4, then a>0 , b < 0.

Depending on which Quadrant (2 or 4) , the point lies, b could be positive or negative.
Hence Statement 1 is insufficient.

Statement 2: a < b

Point (a,b) could in any of the 4 quadrants, so b could be positive or negative. Hence insufficient.

Combining Statement 1 and 2,
Since a< b , we can confirm that point(a,b) is in quadrant 2.
In quadrant 2, b is always positive. So its Sufficient.

Option C is the answer.

Thanks,
Clifin J Francis,
GMAT SME

Kinshook · Answer

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

Since the line k passes through the origin, the equation of the line :
y = mx
Since it passes through point (a, b)
b = ma
m = b/a

y = bx/a

(1) The slope of line k is negative.
m < 0
b/a <0
If a > 0; b<0
But if a<0; b>0
NOT SUFFICIENT

(2) a < b
b/a > 1
a & b both can be positive or both can be negative
NOT SUFFICIENT

(1) + (2)
(1) The slope of line k is negative.
m < 0
b/a <0
Either a is negative or b is negative
(2) a < b
a is negative and b is positive
SUFFICIENT

IMO C

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