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Question Stats: 62% (01:50) correct 38% (01:46) wrong based on 438 sessions

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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
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Joined: 02 Sep 2009
Posts: 58340
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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.

Hope it helps.
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Re: In the xy-plane, the line k passes through the origin and th  [#permalink]

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uzzy12 wrote:
In the xy-plane, the line k passes through the origin and through the point (a,b) where ab doesn't equal zero. Is b positive?

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

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.
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Re: In the xy-plane, the line k passes through the origin and th  [#permalink]

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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.
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Re: In the xy-plane, the line k passes through the origin and th  [#permalink]

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vsaketram wrote:
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.

Every line of the form y = mx passes through the origin, not only y = x and y = -x.
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GMAT 1: 750 Q49 V43 In the xy-plane, the line k passes through the origin and th  [#permalink]

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kilukilam wrote:
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

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 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
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Re: In the xy-plane, the line k passes through the origin and th  [#permalink]

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kilukilam wrote:
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

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
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Re: In the xy-plane, the line k passes through the origin and th  [#permalink]

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kilukilam wrote:
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

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: In this case, b (y-coordinate) is positive. So, the answer to the target question is YES, b is positive

Case b: 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: In this case, b (y-coordinate) is positive. So, the answer to the target question is YES, b is positive

Case b: 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. 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

Cheers,
Brent
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Re: In the xy-plane, the line k passes through the origin and th  [#permalink]

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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.
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Re: In the xy-plane, the line k passes through the origin and th  [#permalink]

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Bunuel wrote:
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.

Hope it helps.

Hey Bunel

When we combine the statements

a<b and -ve slope line

Imagine a case 1 :
(a,b) = (-2,-1) Here a<b and the line has a negative slope, Which means the fourth quadrant.

How can we conclude that together sufficient as the option? Re: In the xy-plane, the line k passes through the origin and th   [#permalink] 10 Jul 2019, 10:46
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