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Re: Coordinate Geometry Data Sufficiency Questions [#permalink]

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12 Mar 2010, 16:30

3

This post received KUDOS

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

Re: Coordinate Geometry Data Sufficiency Questions [#permalink]

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12 Mar 2010, 16:34

1

This post received KUDOS

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

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.

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

both are valid and st2 dont tell anything about quadrants

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.

Answer: C.

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:

II quadrant: point (-3, 2); IV quadrant: point (3, -2).

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

Re: Coordinate Geometry Data Sufficiency Questions [#permalink]

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08 Mar 2012, 13:55

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.

Answer: C.

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?
_________________

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.

Answer: C.

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.
_________________

Re: If ab <> 0 and points (-a,b) and (-b,a) are in the same [#permalink]

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22 Apr 2012, 19:57

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.

gmatclubot

Re: If ab <> 0 and points (-a,b) and (-b,a) are in the same
[#permalink]
22 Apr 2012, 19:57

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