From the given information, (-a,b) and (-b,a) are in the same quadrant, it means two things:
first, -a & -b are both positive or negative (1)
second, b & a are both positive or negative (2)
(1)&(2) suggest that a*b > 0
So if a>0 then (-a,b) will be in Quadrant 1
if a<0 then (-a,b) will be in Quadrant 3.

First premise: x*y> 0 so (-x,y), like (-a,b) can both be in quadrant 1 or 3, however, we do not know whether a & x have the same sign ( + or -). So (-a,b) can be in Q1 if a>0, and (-x,y) can be in Q3 if x<0. => Insuff

Second premise alone : ax>0, we can not determine the sign of y and x, so can not determine the Quadrant the point is in either.
But, combine with the 1st premise, we'll see that a & x have the same sign, so (-a,b) and (-x,y) have the same properties - both stay in Q1 if a>0 and x>0 and in Q3 if a<0 and x<0.

So C. Sorry for my poor drawing.

From the question you should be able to determine that the circumstances are only true when a and b are both positive or both negative and that if they are both positive or both negative, then they are in a different quadrants based on whether they are positive or negative.

So we must ask ourselves are x and y both positive or both negative, and if they are are they the same sign as a or b.

1) answers the first part of the question, but not the second.
2) answer the second part of the question but not the first.

Together they fully answer our prephrase.

Hope this helps.

Thanks,

Jared

Hi all!

if a*b is not equal to 0 then neither a nor b is equal to 0.
if (-a,b) and (-b,a) are in the same quadrant of the (x,y) plane then either a and b both are positive or both are negative.
So, if ( -x,y) in the same quadrant then either x, y and a,b all must be positive or all must be negative.

(1)
if xy > 0 then then either x and y both are positive or both are negative. However, we don't have enough information about a and b.
Therefore, statement (1) ALONE is INSUFFICIENT.

(2)
if ax > 0 then either a and x both are positive or both are negative. However, we don't have information about y.
So, statement (2) ALONE is INSUFFICIENT.

(1) & (2)
if xy > 0 & ax> 0 then points (-a,b) and (-x,y) both are in the same quadrant.
Hence, statements (1) & (2) both are TOGETHER SUFFICIENT. Answer C is correct.

Hope it helps!
Best,
Maciek