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If ab is not 0 and points (-a,b) and (-b,a) are in the same quadrant of the xy-plane, is point (-x,y) in this same quadrant ?

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

IMO C

From what is given, it is clear a and b have the same sign (ie either both are positive or both are negative)

(i) Informs that x and y have the same sign But it is not clear their relationship with a or b and hence can't say which quadrant do they belong.

(ii) This informs that a and x are of the same sign also (ie either both of them are negative or both of them are positive) ii alone does not inform the relationship with y

However, combining them, gives a, b, x and y are all of the same sign. Thus (-x,y) will be in the same quadrant as (-a, b) and (-b, a)

OA ?

OA is C .... what i don't understand is how you determined the portion in red ... _________________

"You have to find it. No one else can find it for you." - Bjorn Borg

From what is given, it is clear a and b have the same sign (ie either both are positive or both are negative) OA is C .... what i don't understand is how you determined the portion in red ...

Since (-a,b) and (-b,a) lie in the same quadrant and x co-ordinates of these two points are -a and -b, this means, -a and -b lie on the same side of x-axis (either positive or negative) and hence both a and b will have the same sign.

If ab is not 0 and points (-a,b) and (-b,a) are in the same quadrant of the xy-plane, is point (-x,y) in this same quadrant ?

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

-----------------------

I got C too, here is my approach.

First, I forced a to be -1 and be to be -2 in order to satisfied (-a,b) and (-b,a) are in same quadrant. so we got (--1,-2) and (--2,-1) --> (1,-2) and (2,-1) which lies in Q4

stmt 1: xy>0, xy must have same sign so if x and y are positive, it makes (-x,y) to be in Q2 --> this case No if x and y are nagative, it make (--x,,-y) to be in Q4 --> this case yes so stmt 1 insuff

stmt 2: ax>o, since we know that a is negative from the number we forced in the beginning, then we know that x is nagative too, however, we don't know about y --> insuff

combine both stmt, we know from 2) that x is negative so y in stmt 1) have to be nagative. So it definitely in Q4 as same as (a,b)

My thought process is stuck on stmt 2, how can I determine what a is given the points? why can't a but a negative therefore making it a positive as a (-a,b)?

Given: ab#0 and points (-a,b) and (-b,a) are in the same quadrant Implies that either both a & b are -ve or both a & b are +ve.

1) either x & y can both be +ve or both be -ve. so we can not decide between 1st and 3rd quadrant 2) we know that a and x have same sign, but unable to determine if they are in same quadrant

together: a,x and y all have same sign => are in same quadrant ( as "b" has same sign as "a")

Let me give this a try. Given: ab is not equal to zero. Points (-a,b) and (-b,a) are in the same quadrant of the xy- plane.

Here are the four quadrants: \(\[ \begin{array}{c|c} II & I \\ \hline III & IV \end{array} \]\)

As the two points are in the same quadrant a and b have the same signs. Lets look at two scenarios. 1. a and b are positive numbers e.g. a = 5 and b = 2 then (-a,b) = (-5,2) and (-b,a) = (-2,5) Here both the points lie in Quadrant II 2. a and b are negative numbers e.g. a = -4 and b = -3 then (-a,b) = (4,-3) and (-b,a) = (3,-4) Here both the points lie in Quadrant IV

The question is asking us if the point (-x,y) in this same quadrant as the above two points? i.e the point (-x,y) would have to be in Quadrant II when a and b are positive or the point (-x,y) would have to be in Quadrant IV when a and b are negative. This would mean that x and y would have to share the same sign as a and b to belong in the same quadrant.

Lets look at statement 1: xy is greater than 0 => x and y are both positive or x and y are both negative. => This atleast tells us that the point in either in Quadrant II or IV. => We don't have any relationship of x or y to a or b. Hence Not Sufficient.

Lets look at statement 2: ax is greater than 0 => a and x are both positive or a and x are both negative. => This tells us that x shares the same sign as a (and b) but we still don't know anything about y. Hence Not sufficient.

Looking at them together; 1 tells us x and y have the same sign. 2 tells us x as the same sign as a and b. Which means y as the same sign as a and b. => x and y are positive when a and b are positive or x and y are negative when a and b are negative. Hence, the point (-x,y) lies on the same quadrant as the points (-a,b) and (-b,a).

Ans- C A different approach. The points (-a,b) and (-b,a) are in the same quadrant. There is no harm in thinking that both of these are same point and we are just representing it with different variables. If these are same point then -a = -b and b = a which means that a and be both should have the same sign.

Stmt 1. XY > 0 tells that either X and Y are both +ve or both -ve.we are not sure So we can not tell where is (-x,y) as it can be 1st or 3rd quadrant.

stmt 2. ax > 0 if a = +ve means x = +ve if a = -ve means x = -ve no suff.

Combine, we can say that a and y have the same sign and the sign of X will be same as that of Y (which we already concluded). Hence suff.

Hi all thanks for the explanations but i still do not get the questiounderstand .. Let me try and explain where my head is it

1) -a and b to be in the same quadrent - could it be that\(-a = -3\) and \(b = 3\) Is this what you mean by both points having the same sign ? Can some one explain with a diagram please..

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