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# If ab <> 0 and points (-a,b) and (-b,a) are in the same

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If ab <> 0 and points (-a,b) and (-b,a) are in the same [#permalink]  12 Mar 2010, 14:35
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1) If ab <> 0 and points (-a,b) and (-b,a) are in the same quadrant of the xy-plane, is point (-x,y) in the same quadrant?

(1) xy > 0
(2) ax > 0

2) 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
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  12 Mar 2010, 15:30
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ahirjoy wrote:
1) If ab <> 0 and points (-a,b) and (-b,a) are in the same quadrant of the xy-plane, is point (-x,y) in the same quadrant?

(1) xy > 0
(2) ax > 0

given a!=0 and b!=0 ... a could be positive or negative and b could be positive or negative
so 4 cases and the respective points mentioned ...
a>0, b>0 ... points would be (-a,b) and (-b,a) and these both lie in quadrant II. GOOD
a>0, b<0 ... points would be (-a,-b) and (b,a) and these both lie in different quadrants. NOT GOOD
a<0, b>0 ... points would be (a,b) and (-b,-a) and these both lie in different quadrants. NOT GOOD
a<0, b<0 ... points would be (a,-b) and (b,-a) and these both lie in quadrant IV. GOOD
so if a>0, b>0 then the given points lie in Q II
so if a<0, b<0 then the given points lie in Q IV

st 1) xy>0
both x,y > 0 .. point (x,y) is in Q I
both x,y < 0 .. point (x,y) is in Q IV
not sufficient
st 2) ax > 0
both x,a > 0 .. point (x,y) could be in Q I or QIV
both x,a < 0 .. point (x,y) could be in Q II or QIII
not sufficient

combining

a,x,y > 0 ... points would be (-a,b) and (-b,a) and these both lie in quadrant II. and point (x,y) would be in Q I
a,x,y < 0 ... points would be (-a,b) and (-b,a) and these both lie in quadrant IV. and point (x,y) would be in Q III

in either case, we can say (x,y) is not in the same quadrant

C
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  12 Mar 2010, 15:34
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ahirjoy wrote:
2) 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

a!=0 and b!=0 .. is b>0
st 1) slope is negative .. line will be in Quadrant II(b is positive) and Quadrant IV(b is negative)
not sufficient
st 2) a<b
this could happen in Q I, QII, Q III
not sufficient

combining
in Q IV. a is positive and b is negative, so a is always > b ..
so the point (a,b) can only be in Q II .. and b is positive

C
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  30 May 2010, 11:56
st 2) a<b
this could happen in Q I, QII, Q III
not sufficient

Why not a<b in Q IV? For example, a=-3 and b=-2 a< b and in Q IV...did I miss any thing? Thought answer is E
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  30 May 2010, 12:23
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tochiru wrote:
st 2) a<b
this could happen in Q I, QII, Q III
not sufficient

Why not a<b in Q IV? For example, a=-3 and b=-2 a< b and in Q IV...did I miss any thing? Thought answer is E

You are right: if $$a<b$$, point $$(a, b)$$ can be in any quadrant. But the answer to this question is still C.

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) 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.

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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  30 May 2010, 19:08
Sorry..still didnt get.

How can we conclude that a and b have opp signs and hence in Q 4?

St1 tells whether q1 or Q4, but st2 dont tell anything about quadrants..no?
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  31 May 2010, 03:18
Bunuel, thanks for the explanation.
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  02 Jun 2010, 20:07

How can we conclude that a and b have opp signs (based on a<b )and hence in Q 4?

Q1 case-- (a,b) = 1,2
Q4 case-- (a,b) = -3,-2

St1 tells whether point is in q1 or Q4

I am not what i am missing here..
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  03 Jun 2010, 04:39
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tochiru wrote:

How can we conclude that a and b have opp signs (based on a<b )and hence in Q 4?

Q1 case-- (a,b) = 1,2
Q4 case-- (a,b) = -3,-2

St1 tells whether point is in q1 or Q4

I am not what i am missing here..

(1) 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).

In II quadrant x-s are negative and y-s are positive, hence if point $$(a,b)$$ is in this quadrant, $$b$$ (y coordinate of the point) is positive;
In IV quadrant x-s are positive and y-s are negative, hence if point $$(a,b)$$ is in this quadrant, $$b$$ (y coordinate of the point) is negative;

The above means that x and y coordinates of the point $$(a,b)$$ have opposite sign (if $$a$$ positive then b negative and vise-versa).

So, $$b$$ can be positive or negative. Not sufficient.

(2) $$a<b$$, not sufficient by itself.

(1)+(2) $$a<b$$ --> $$a$$ is less than $$b$$, as they have opposite signs, then $$a$$ must be negative and $$b$$ positive (point lies in the second quadrant). Sufficient.

The examples you provide are not valid:

Q1 case-- (a,b) = 1,2 - point $$(a,b)$$ is in II quadrant not in I.
Q4 case-- (a,b) = -3,-2 - point $$(-3, -2)$$ is in III quadrant not in IV.

Proper examples would b:

For more on this issue please check Coordinate Geometry chapter of Math Book (link in my signature).

Hope it helps.
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  08 Mar 2012, 12:55
Bunuel wrote:
You are right: if $$a<b$$, point $$(a, b)$$ can be in any quadrant. But the answer to this question is still C.

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) 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.

i have a question: if ab is not equal to 0, then it means that a & b are either both +ve or both -ve [quadrant I or III], and that b/a=m can't be equal to 0 either. with this info, how can (a,b) be in any quadrant and not just I or III?
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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]  08 Mar 2012, 13:00
Expert's post
MBAhereIcome wrote:
Bunuel wrote:
You are right: if $$a<b$$, point $$(a, b)$$ can be in any quadrant. But the answer to this question is still C.

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) 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.

i have a question: if ab is not equal to 0, then it means that a & b are either both +ve or both -ve [quadrant I or III], and that b/a=m can't be equal to 0 either. with this info, how can (a,b) be in any quadrant and not just I or III?

$$ab\neq{0}$$ just means that neither $$a$$ nor $$b$$ equal zero, but we cannot say anything about their sings.
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Re: If ab <> 0 and points (-a,b) and (-b,a) are in the same [#permalink]  22 Apr 2012, 18:10
... hmmm why do we need to consider line and all.. my approach:

as (-a,b) and (-b,a) are in same quad that mean -a and -b are same sigh similarly b and a are same sign. Thus a and b are either both +ve or both -ve.

from stat1 - we just know about x and y that these are either both +ve and both -ve and based on that we cant say they are in same quad as a,b

stat 2 - a and x are both +ve or both -ve but we donno about x and y relation...

combining stat 1 and stat2
x,y,a,b are all +ve or x,y,a,b are all -ve thus (x,y) and (a,b) in same quad.
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Re: If ab <> 0 and points (-a,b) and (-b,a) are in the same [#permalink]  22 Apr 2012, 18:57
Thanks khushboochhabra, that is definitely an easier way to look at the problem... For those still in doubt, drawing a quick table with possible +/- combinations will help confirm that a & b need to be the same sign.
Re: If ab <> 0 and points (-a,b) and (-b,a) are in the same   [#permalink] 22 Apr 2012, 18:57
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