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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
(1) xy > 0
(2) ax > 0
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15 May 2010, 07:31
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If ab different from 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?
The fact that points \((-a,b)\) and \((-b,a)\) are in the same quadrant means that \(a\) and \(b\) have the same sign. These points can be either in II quadrant, in case \(a\) and \(b\) are both positive, as \((-a,b)=(-,+)=(-b,a)\) OR in IV quadrant, in case they are both negative, as \((-a,b)=(+,-)=(-b,a)\) ("=" sign means here "in the same quadrant").
Now the point \((-x,y)\) will be in the same quadrant if \(x\) has the same sign as \(a\) (or which is the same with \(b\)) AND \(y\) has the same sign as \(a\) (or which is the same with \(b\)). Or in other words if all four: \(a\), \(b\), \(x\), and \(y\) have the same sign.
Note that, only knowing that \(x\) and \(y\) have the same sign won't be sufficient (meaning that \(x\) and \(y\) must have the same sign but their sign must also match with the sign of \(a\) and \(b\)).
(1) \(xy>0\) --> \(x\) and \(y\) have the same sign. Not sufficient.
(2) \(ax>0\) --> \(a\) and \(x\) have the same sign. But we know nothing about \(y\), hence not sufficient.
(1)+(2) \(x\) and \(y\) have the same sign AND \(a\) and \(x\) have the same sign, hence all four \(a\), \(b\), \(x\), and \(y\) have the same sign. Thus point \((-x,y)\) is in the same quadrant as points \((-a,b)\) and \((-b,a)\). Sufficient.
Answer: C.
For more in this topic check coordinate geometry chapter of math book: math-coordinate-geometry-87652.html
Hope it helps.
The fact that points \((-a,b)\) and \((-b,a)\) are in the same quadrant means that \(a\) and \(b\) have the same sign. These points can be either in II quadrant, in case \(a\) and \(b\) are both positive, as \((-a,b)=(-,+)=(-b,a)\) OR in IV quadrant, in case they are both negative, as \((-a,b)=(+,-)=(-b,a)\) ("=" sign means here "in the same quadrant").
Now the point \((-x,y)\) will be in the same quadrant if \(x\) has the same sign as \(a\) (or which is the same with \(b\)) AND \(y\) has the same sign as \(a\) (or which is the same with \(b\)). Or in other words if all four: \(a\), \(b\), \(x\), and \(y\) have the same sign.
Note that, only knowing that \(x\) and \(y\) have the same sign won't be sufficient (meaning that \(x\) and \(y\) must have the same sign but their sign must also match with the sign of \(a\) and \(b\)).
(1) \(xy>0\) --> \(x\) and \(y\) have the same sign. Not sufficient.
(2) \(ax>0\) --> \(a\) and \(x\) have the same sign. But we know nothing about \(y\), hence not sufficient.
(1)+(2) \(x\) and \(y\) have the same sign AND \(a\) and \(x\) have the same sign, hence all four \(a\), \(b\), \(x\), and \(y\) have the same sign. Thus point \((-x,y)\) is in the same quadrant as points \((-a,b)\) and \((-b,a)\). Sufficient.
Answer: C.
For more in this topic check coordinate geometry chapter of math book: math-coordinate-geometry-87652.html
Hope it helps.
04 Jan 2012, 12:41
Kudos
Bookmarks
Hi, there! I'm happy to help with this. 
First, a quick review of quadrants: what defines the quadrants are the +/- signs of x and y
1) In Quadrant I, x > 0 and y > 0
2) In Quadrant II, x < 0 and y > 0
3) In Quadrant III, x < 0 and y < 0
4) In Quadrant VI, x > 0 and y < 0
If (-a, b) and (-b, a) are in the same quadrant, that means that the x-coordinates have the same sign, and also the y-coordinates have the same sign. Look at the y-coordinates --- if the two points are in the same quadrant, a & b have the same sign. They either could both be positive (in which case, the points would be in Quadrant II) or they could both be negative (in which case, the points would be in Quadrant IV).
Now, the question is: (-x, y) in the same quadrant as these two points?
(1) Statement 1: xy > 0
This tells us that x and y have the same sign --- both positive or both negative. Now, we know a & b have the same sign, and x & y have the same sign, but there's two possibilities for each, so we don't know whether a & b & x & y all have the same sign. This is insufficient.
(2) Statement 2: ax > 0
This, by itself, tells us that a and x have the same sign -- with this alone, we know that a & b & x all have the same sign, but we have zeor information about y. This too is insufficient.
Combined (1) & (2)
Prompt tells us a & b have the same sign. Statement #1 tells us x & y have the same sign. Statement #2 tells us x & a have the same sign. Put it all together --> we now know that x & y & a & b all have the same sign. Therefore, (-x, y) will have the same sign x- & y-coordinates as (-a, b) & (-b, a), and therefore all will be in the same quadrant. Combined statements are sufficient.
Answer = C
Here's another coordinate plane practice question just for practice.
https://gmat.magoosh.com/questions/1028
Does all that make sense? Please let me know if you have any additional questions.
Mike
First, a quick review of quadrants: what defines the quadrants are the +/- signs of x and y
1) In Quadrant I, x > 0 and y > 0
2) In Quadrant II, x < 0 and y > 0
3) In Quadrant III, x < 0 and y < 0
4) In Quadrant VI, x > 0 and y < 0
If (-a, b) and (-b, a) are in the same quadrant, that means that the x-coordinates have the same sign, and also the y-coordinates have the same sign. Look at the y-coordinates --- if the two points are in the same quadrant, a & b have the same sign. They either could both be positive (in which case, the points would be in Quadrant II) or they could both be negative (in which case, the points would be in Quadrant IV).
Now, the question is: (-x, y) in the same quadrant as these two points?
(1) Statement 1: xy > 0
This tells us that x and y have the same sign --- both positive or both negative. Now, we know a & b have the same sign, and x & y have the same sign, but there's two possibilities for each, so we don't know whether a & b & x & y all have the same sign. This is insufficient.
(2) Statement 2: ax > 0
This, by itself, tells us that a and x have the same sign -- with this alone, we know that a & b & x all have the same sign, but we have zeor information about y. This too is insufficient.
Combined (1) & (2)
Prompt tells us a & b have the same sign. Statement #1 tells us x & y have the same sign. Statement #2 tells us x & a have the same sign. Put it all together --> we now know that x & y & a & b all have the same sign. Therefore, (-x, y) will have the same sign x- & y-coordinates as (-a, b) & (-b, a), and therefore all will be in the same quadrant. Combined statements are sufficient.
Answer = C
Here's another coordinate plane practice question just for practice.
https://gmat.magoosh.com/questions/1028
Does all that make sense? Please let me know if you have any additional questions.
Mike
