first let's examine how (-a, b) and (-b, a) are in the same quadrant. In order for both these points to be in the same quadrant, the following condition must be met:
(the sign of -a = the sign of -b) AND (the sign of b = the sign of a)
here, I am basically comparing the x-axis values and y-axis values separately. in order to meet this condition, both and b must have the same sign. Either both must be positive or both must be negative.
-if both a and b are positive, then the two points will lie in the 2nd quadrant (negative x, positive y).
-if both a and b are negative, then the two points will lie in the 4th quadrant (positive x, negative y).
now we need to establish which quadrant does point (-x, y) lie in using the statements.
statement 1: xy > 0
either
x > 0, y > 0, so point (-x, y) will lie in the 2nd quadrant,
or
x < 0, y < 0, so point (-x, y) will lie in the 4th quadrant.
although the two earlier points also lie in one of these 2 quadrants, we still don’t know exactly which one. Hence insufficient.
statement 2: ax > 0
either
a > 0, x > 0, so points (-a, b) and (-b, a) will lie in the 2nd quadrant, but we have no info on y,
or
a < 0, x < 0, so points (-a, b) and (-b, a) will lie in the 4th quadrant, but we have no info on y.
in either case, we don’t know anything about y, hence cannot conclude anything. insufficient.
both statements together:
if x>0, y>0, then from S2 a>0 and hence b>0. this means all points are in the 2nd quadrant.
if x<0, y<0, then from S2 a<0 and hence b<0. this means all points are in the 4th quadrant.
in both cases we have (-x, y), (-a, b) and (-b, a) within the same quadrant. this answers the question stem as a definite YES. Sufficient.
Choose C.
-BM-
