I think it's easier to explain if we pick some numbers.
Let's be a=3 and b=2 so points (-a,b) and (-b,a) are (-3,2) and (-2,3) both in quadrant II.

1. xy >0
Let's be x=4 and y=5, so (-4,5) is in quadrant II.
But x could be -4 and y could be -5, so (4,-5) is in quadrant IV.
NOT SUFFICIENT

2. ax>0
For a = 3 and x=4, if y=5, (-4,5) is in quadrant II but if y=-5, point (-4,-5) is in quadrant III.
NOT SUFFICIENT.

Both 1. and 2.
If is a=3, x=4 AND y=5, the point (-4,5) is be in quadrant II.
SUFFICIENT -> the answer is C

Hope this helps. You can try other numbers (a=-2,b=-3) but the result is similar.

If is a=3, x=4 AND y=5, the point (-4,5) is be in quadrant II.
SUFFICIENT -> the answer is C

how can you assume y to be a positive number? from the two stmnts, x and y have the same sign, which can be negative Or positive.
If both are positive, then the points lie in quadrant II. However, if both are negative, the points lie in quadrand IV

Am I missing something here?

vkb16 wrote:

If is a=3, x=4 AND y=5, the point (-4,5) is be in quadrant II.
SUFFICIENT -> the answer is C

how can you assume y to be a positive number? from the two stmnts, x and y have the same sign, which can be negative Or positive.
If both are positive, then the points lie in quadrant II. However, if both are negative, the points lie in quadrand IV

Am I missing something here?

For (-a,b) and (-b,a) to be in the same quadrant, a and b must be the same sign.

For (-x, y) to be in the same quadrant as (-a,b) and (-b,a), x,y,a must be the same sign or x, y, b must be the same sign.

1> xy>0 -> tells us both x and y are the same sign but that doesn't tell us whether this is the same sign as a&b -> insuff

2> ax>0 -> both a and x are same sign; this doesn't tell us anything about the sign of y -> insuff

1 and 2. x, y and a are the same sign and since a and b are the same sign a, b, x, y are all same sign hence suff.

hope that helps.