If ab0 and points (-a, b) and (-b, a) are in the same quadrant of the

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.
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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
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(-a, b) and (-b, a) are in the same quadrant.
Is the point (-x, y) in the same quadrant as point (-a, b)?

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

From the information that (-a, b) and (-b, a) are in the same quadrant, it can be determined that
(-a, b) is in either quadrant II or quadrant IV. If (x, y) and (a, b) are in the same exact quadrant, they will have the same sign and (-x, y) will be in the same quadrant as (-a, b)'s.

(1) xy > 0

(x, y) is in quadrant I or quadrant III.
(-x, y) is in quadrant II or quadrant IV.
No further information is provided about (-a, b).

(2) ax > 0

Point x in (x, y) has the same sign as does point a in (a, b). Since a and b have the same sign, x, a and b have the same sign.

But the sign of point x could be different from, or the same as, the sign of point y. The condition that (x, y) and (a, b) have the same sign, and therefore that (-x, y) and (-a, b) are in the same quadrant, is possible but uncertain.

Combined analysis:

x has the same sign as y
x has the same sign as a and b
x, y, a and b all have the same sign.

This means (x, y) and (a, b) are in the same quadrant. (-x, y) and (-a, b) are in the same quadrant.


[xyab+xdj]

Thanks everyone. But I am still getting confused between x, y a and b. Are we saying x and y as cordinates and a and b as points i.e. x(-a,b) and y(-b,a)?
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We have 3 points with coordinates (-a,b), (-b,a) and (-x, y).

Also, check Coordinate Geometry chapter of Math Book: math-coordinate-geometry-87652.html

Hope it's clear.
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(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.

Bunuel, you are saying that (1)+(2) tells us that ALL a, b, x, and y have the same sign
Here's my doubt:
statements (1)+(2) give us info ONLY about the signs of a, x, and y.
You are telling that if "a, x, and y all have the SAME sign then b also has the same sign as a, x, and y."
How could you a say that because b does not form part of any of the statements (1) or (2)
So, what I mean to say is that b can be +ve or -ve irrespective of what signs a, x, and y take.
Please clear my doubt Bunuel.

The fact that points \((-a,b)\) and \((-b,a)\) are in the same quadrant means that \(a\) and \(b\) have the same sign.
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Hey Bunuel just asking a relevant doubt. Does ( -b,-a) or (-a,-b) lies in the same quadrant as (a,b) ?

Do you mean generally? If yes, then:

(a, b) and (-a, -b) will never be in the same quadrant.

(a, b) and (-b, -a) will be in the same quadrant if a is positive and b is negative, in this case (a, b) = (+, -) and (-b, -a) = (+, -) OR when a is negative and b is positive, in this case (a, b) = (-, +) and (-b, -a) = (-, +).
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ab≠0 means neither a nor b can be 0.

(-a, b) and (-b, a) are in the same quadrant of the xy-plane - In a quadrant, all x co-ordinates have the same sign and all y co-ordinates have the same sign e.g. all x coordinates are positive and all y coordinates are also positive in the first quadrant. Similarly, all x coordinates are negative and all y coordinates are positive in the second quadrant and so on...

So sign of -a and sign of -b should be the same. This means a and b must have the same signs. So either a and b are both positive such that (-a, b) is in second quadrant or both negative such that (-a, b) is in fourth quadrant.

Ques: is point (-x, y) in the same quadrant?

For (-x, y) to be in the same quadrant as (-a, b) and (-b, a), signs of x and y should be same and they should be the same as the sign of a and b.
If a and b are positive, (-x, y) should lie in the second quadrant so x and y should be positive too.
If a and b are negative, (-x, y) should lie in the fourth quadrant so x and y should be negative too.

(1) xy > 0

'x' and 'y' have the same sign but is the sign same as that of 'a' and 'b'? We don't know.

(2) ax > 0

'a' and 'x' have the same sign but is the sign of 'y' same too? We don't know.

Both together we know that 'x' and 'y' have the same sign and their sign is the same as the sign of 'a' too. Sufficient.

Answer (C)
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If ab does not equal zero and points (-a,b) and (-b,a) are in the same quadrant of the xy-plane,
here's how to interpret ^^ this:
you should immediately make the following association: QUADRANTS --> SIGNS
so ... the x- and y-coordinates of both points have to have the same signs.
this means that
-a and -b have the same sign (from the x coordinates)
and
b and a have the same sign (from the y coordinates)
these are of course the same statement; both are equivalent to saying that a and b have the same sign.
therefore, that's all we know from the problem statement: a and b are nonzero and have the same sign. note that we do not know which sign that is!
is point (-x,y) in the same quadrant?
^^ we need to know two things:
(1) whether -x has the same sign as both -a and -b --> whether x has the same sign as a and b (from the first coordinate)
(2) whether y has the same sign as a and b (from the second coordinate)

Statement (1): xy>0
^^ this means x and y have the same sign as EACH OTHER, but we don't know how it relates to a or b.
insufficient

Statement (2): ax>0
^^ this means x has the same sign as a (and therefore b), but we don't have any information about y.
insufficient

together: x has the right sign, and y has the same sign as x does. therefore, both of them have the correct sign!
answer = c

**THE ENTIRE POINT OF DATA SUFFICIENCY is to reward test takers who actually focus on the goal of the problem, and to punish those who don't. if you begin your work thinking "i need to find a specific quadrant", you've already gotten the problem wrong, BEFORE YOU EVEN START WORKING ON IT.
this is, in fact, the entire reason why the DS format was invented.
a multiple-choice question can't punish you for finding too much information. (if you can find the information... well, good for you.)
on the other hand, DS problems can... and they do.
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One of the Best Explanations out there! (collected)
“If ab does not equal zero and points (-a,b) and (-b,a) are in the same quadrant of the xy-plane,”
^^ here's how to interpret this:
you should immediately make the following association: QUADRANTS --> SIGNS
so ... the x- and y-coordinates of both points have to have the same signs.
this means that
-a and -b have the same sign (from the x coordinates)
and
b and a have the same sign (from the y coordinates)

^^ these are of course the same statement; both are equivalent to saying that a and b have the same sign.
therefore, that's all we know from the problem statement: a and b are nonzero and have the same sign.
note that we do not know which sign that is!

“is point (-x,y) in the same quadrant?”
^^ we need to know two things:
1/ whether -x has the same sign as both -a and -b --> whether x has the same sign as a and b (from the first coordinate)
2/ whether y has the same sign as a and b (from the second coordinate)

Statement (1): xy>0
this means x and y have the same sign as EACH OTHER, but we don't know how it relates to a or b.
insufficient

Statement (2): ax>0
this means x has the same sign as a (and therefore b), but we don't have any information about y.
insufficient

together: x has the right sign, and y has the same sign as x does. therefore, both of them have the correct sign!
answer = c
sweetness
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ThatDudeKnows BrentGMATPrepNow ScottTargetTestPrep could you kindly share your insights on this question please? Is - sign in (-a, b) and (-b, a) relevant here at all? Thanks for your time in advanced.

Useful property: -k = (-1)(k)
So you can read -b as (-1)(b), and -a as (-1)(a)
Does that help?

please advice when we say (-a,b) don't we mean -a is x axis and b is y axis which would mean it is 2 quadrant. Same for (-b,a). If so (-x,y) should mean they are also in quadrant 2. I am confused because I was taught we always write in sequence of x axis first and y axis later

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-a is not necessarily negative and b is not necessarily positive. So, (-a,b) is not necessarily in the II quadrant. For example, if a = -1 and b = -1, then (-a,b) = (1, -1), so it's in the IV quadrant, not in II.
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Thanks for reply but again then ab not equal to zero will can have two conditions both+ve, both-ve or one is +ve and one is -ve

Not sure what you imply there but ab ≠ 0 means that neither a nor be is 0, so each of the a and be can be positive or negative.
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Thanks again,guess i need to to take a break.

I was implying as below

-ab.........0.........+ab

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