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

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

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21 May 2010, 11:17
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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-ab-0-and-a-b-and-b-a-are-in-the-same-quadrant-is-126039.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 13 Oct 2013, 15:52, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
Manager
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Re: Coordinate Geometry [#permalink]

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21 May 2010, 11:40
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It's much easier to work toward an explanation when you know the right answer, heh.

From the first part of the question, we can be sure that the sign (+/-) of x has to equal the sign of y since (-x,y) and (-y,x) are in the same quadrant.

(1) Tells us that whatever the signs of s and t are, they have to be same (otherwise > 0 doesn't hold). But this means (-s,t) could be in one of two quadrants. (A is out, thus so is D)

(2) Tells us that the sign of t and x must be the same. Taken alone, there's not much you can do because you don't know anything about s. (B is out)

Taken together: Since sign of s = t and sign of x = t: sign of t = s = x, and furthermore sign of y = x = t = s (from the question)
All the signs of the variables are the same. So (-s,t) will be in the same quadrant of (-x,y), and therefore in the same quadrant of (-y, x) (as would (-t, s). --- Answer: C

Without the OA I probably would taken much longer to work that out and maybe gotten wrong, so thanks!
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Re: Coordinate Geometry [#permalink]

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21 May 2010, 16:13
If xy does not equal 0 and points (-x,y) ans (-y,x) are in the same quadrant of the xy-plane, is point (-s, t) in the same quadrant?

(1) st > 0
(2) xt > 0

Answer is C , However I prefer a different approach to solve this problem.

Use actual number to solve the probelm.

Given premise claim that sign of x and y must be same.
So consider X= 5 and Y = 4. (-5,-4) and (-4,5) are in II quadrant.
Consider X = -7 and Y = -8, (7,-8) and (8,-7) are in IV quadrant.

Statement 1 : Sign of s and t are same.
Consider S = 2 , t = 3 , (-2, 3) is in II quadrant
Consider S = -2, t = -3 ,( 2, -3) is in IV quadrant.

Statement 2: Sign of x and t are same. When X is + , t must be +. When X in -, t must be -.

Togather, we can say (-x,y) , (-y,x) and (-s, t) are in same quadrant : II or IV
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Re: If xy does not equal 0 and points (-x,y) ans (-y,x) are in [#permalink]

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13 Oct 2013, 15:44
marcusaurelius wrote:
If xy does not equal 0 and points (-x,y) ans (-y,x) are in the same quadrant of the xy-plane, is point (-s, t) in the same quadrant?

(1) st > 0
(2) xt > 0

OA= C

Please don't leave OA naked. Then it makes it hard for us to solve such a nice problem in real conditions.
Cheers,
J
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Re: If xy does not equal 0 and points (-x,y) ans (-y,x) are in [#permalink]

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13 Oct 2013, 15:53
Expert's post
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BOOKMARKED
jlgdr wrote:
marcusaurelius wrote:
If xy does not equal 0 and points (-x,y) ans (-y,x) are in the same quadrant of the xy-plane, is point (-s, t) in the same quadrant?

(1) st > 0
(2) xt > 0

OA= C

Please don't leave OA naked. Then it makes it hard for us to solve such a nice problem in real conditions.
Cheers,
J

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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-ab-0-and-a-b-and-b-a-are-in-the-same-quadrant-is-126039.html
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Re: If xy does not equal 0 and points (-x,y) ans (-y,x) are in   [#permalink] 13 Oct 2013, 15:53
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# If ab≠0 and points (-a,b) and (-b,a) are in the same quadran

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