Are you sure that you are wrong?

A(-a,b) B(-b,a) lie in same quadrant and thus it means a and b are of same sign. they can be ++ or --
thus -a,b are of opposite signs and also -b,a are of opposite signs.
it means A,B are either in 2nd quadrant or in 4th quadrant
to find if C(-x,y) lies in same quadrant as A,B
a)xy>0
it means x,y are of same sign. thus -x,y are of opposite sign
Thus C lies either in 2nd or 4th quadrant
But we have possibility that A,B lie in 2nd while C lies in 4th
Or another possibility that A,B,C lie in 2nd. Thus insufficient
b)ax>0
it means a,x are of same sign. let a>0, A,B lie in 2nd quadrant
Also then x>0, or -x<0 and no information on C, thus C can lie either in 2nd quadrant or in 3rd quadrant
Insufficient
both together) C either in 2nd or in 4th. but x,y of same sign, thus y>0 and thus C in 2nd quadrant. Also A,B in 2nd only
Sufficient
IMO C

@Cans Awesome explanation. Looks like i have to improve my comprehension of the question

I was assuming the question to be is(-x,y)in the same quadrant as (-y,x). Thanks for the expln!!