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If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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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

Originally posted by jitendra on 15 May 2010, 04:37.
Last edited by Bunuel on 31 Jul 2014, 01:52, edited 3 times in total.
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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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.

For more in this topic check coordinate geometry chapter of math book: math-coordinate-geometry-87652.html

Hope it helps.
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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I took my first GmatPrep today after studying Quant for a month(Working on a 3month plan suggested by gmatclub experts). I haven't touched Verbal yet and my score was 660 (Q49V31) although i was a little disturbed about my Verbal score since i expected better, i was pretty surprised how getting 13 questions wrong in Quant got me to 49. But since GMAT is adaptive i guessed its possible.
Anyways, i reworked the incorrect questions after the exam and cracked a few of them, however there are a few others that just stumped me completely even after giving them a 2nd shot.

1. If ab!=0 and point (-a,b) and (-b,a) are in the same quadrant ,does point (-x,y) lie in this quadrant?
i) xy>0
ii) ax>0

There are a few others coming up..Please let me know if I made a rookie mistake by posting these here when it should be in some other forum category, I searched a lot couldn't really find any other suitable place. Thanks Originally posted by ijoshi on 07 Dec 2011, 10:43.
Last edited by ijoshi on 07 Dec 2011, 17:28, edited 2 times in total.
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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1
(-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

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]

Originally posted by Study1 on 18 Dec 2011, 15:48.
Last edited by Study1 on 31 Jan 2012, 15:26, edited 3 times in total.
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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11
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.

Here's another coordinate plane practice question just for practice.

http://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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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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1
Lets rephrase the stem first. For (-a,b) and (-b, a) to lie in same quadrant, both are either positive or negative.

1. xy>0, which means both are either positive or negative. Say a and b are positive, so they lie in IV. But xy could be ++ or --, causing it to lie in II or IV. Insufficient.

2. ax>0. which means positive or negative. What about y? No data on y causes this statement to be insufficient.

Together, means that a, x and y have same signs, therefore same quadrants. Sufficient - C.
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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

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enigma123 wrote:
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)?

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

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Yes Bunuel - got it now. Thanks.
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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1
jitendra wrote:
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

A table will help ..

a b -a,b -b,a
+ + 2nd 2nd
+ - 3rd 1st
- + 1st 3rd
- - 4th 4th

you gotta know following:
+,+ >> 1st
-,+ >> 2nd
-,- >> 3rd
+,- >> 4th

this tells us (-a, b) and (-b, a) are either in 2nd quadrant or in 4th quadrant ..

1.) xy>0 means both have same sign and -x,y could be in 2nd or 4th quadrant .. its possible that -x,y is in 4th quadrant and (-a, b) and (-b, a) in 2nd and vice-a-versa .. hence insufficient

2.) ax>0 .. no info about y ... not sufficient

1+2 >> a and x both +ve 2nd qadrant
both negative, 4th quatrant .. hence -x,y and the points given in question will be in same quadrant .. C answer
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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

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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.

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

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mikemcgarry wrote:
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).

Can someone please provide insights in the above colored part.
I'm not sure if I would be able to deduce it under timed conditions. I know, this can be proved by taking hypothetical coordinates and see the behavior. However, I would like to understand it conceptually.

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

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ab≠0 and points (-a,b) and (-b,a) are in the same quadrant → tells me that a and b are both + or -

(1) xy>0 → tells me that x and y are both + or -. Not suffient

(2) ax>0 → tells me that a, b and x are all + or -. Not suffient

(1)+(2) enabled me to answer the question: C
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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Bunuel wrote:
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.

For more in this topic check coordinate geometry chapter of math book: math-coordinate-geometry-87652.html

Hope it helps.

Hey Bunuel just asking a relevant doubt. Does ( -b,-a) or (-a,-b) lies in the same quadrant as (a,b) ?
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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Sidhrt wrote:
Bunuel wrote:
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.

For more in this topic check coordinate geometry chapter of math book: math-coordinate-geometry-87652.html

Hope it helps.

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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Re: If ab ≠ 0 and points (–a, b) and (–b, a) are in the same quadrant ..  [#permalink]

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From the question stem, we can clearly tell that (–a, b) and (–b, a) are in the same quadrant
if both a,b are positive or negative.
We are asked to find if the point(-x,y) is also in the same quadrant

1. if xy > 0 , both x and y are postive, or both their values are negative are two options available.
When x and y are positive, if a and b are negative, they will not be in same quadrant.
But, if x and y are postive and a and b are also positive, these points will be in the same quadrant
Hence, insufficient.

2. ax > 0, both a and x are positive or both of them are negative.
When a and x are postive, b and y can be both negative or positive.
In one of the cases, we will have a YES, , for the point(-x,y) being in the same quadrant.
whereas in the other case, we will have a NO, for the point(-x,y) being in the same quadrant.
Hence insufficient.

But on combining the two, either a,b,x,y are all negative or all positive and clearly lie in the same quadrant
Sufficient(Option C)
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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jitendra wrote:
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

we can infer that a and b are the same sign
pick numbers noun
a=b=1, the two points are at quater 2
x=1, y=1, it is in quater 1
x=-1, y=-1, the point is quater 2.
condition 1 is not enough

we can do this way. pick specific numbers for easy understanding.
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Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr  [#permalink]

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jitendra wrote:
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

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.

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

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Hi mikemcgarry,

One thing regarding this question has been bugging my mind, could you please help me with it?

For some reason, I cannot get the notion out of my head that the quadrants always have to be in the following formats:

However, if we take the signs of both "a" and "b" to be negative then couldn't it look different?
Lets say: a = -2 and b = -3 then we know that these point falls on the 3rd quadrant but while putting the values in (-a, b) and (-b, a) they would come to be as (2, -3) and (3, -2) which are the standard form of the 4th quadrant (x, -y) isn't it?
Now I know while plotting the values they would fall on the 3rd quadrant but its the confidence that I am lacking when tackling a similar question.

Thank You,
Dablu Re: If ab≠0 and points (-a, b) and (-b, a) are in the same quadr   [#permalink] 25 Nov 2019, 05:35

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