since (-a,b) and (-b,a) are on the same quadrant then tank = (a-b)/(-b+a) = 1 so pts. are on 1st or 3rd quadrant (since tank is +ve)
a) xy > 0 pts. (x,y) are on 1st or 3rd quadrants --insuff--
b) ax > 0 pts. (a,x) are on 1st or 3rd quadrants --insuff--
a), b) if we multiply ax>0 by 1/xy we have a/y>0 or ay > 0
so we get followimg cases
i)a > 0, y > 0, -x < 0 so (-x,y) -->(2nd quadrant)
ii)a < 0, y < 0, -x > 0 so (-x,y) -->(4th quadrant)
so (-x,y) is not in the same quadrant with other 2 pts.
The quadrant is determined by the sign of x & y coordinates.
(+,+) - 1st
(-,+) - 2nd
(-,-) - 3rd
(+,-) - 4th
Now question -
(-a, b) & (-b, a) are in same quadrant
=> a and b have same sign,(either both are +ve or -ve)
(-x, y) will be in the same quardant if
both x y have same sign and also sign of x & y is same as that of a & b
1> tells x,y are of same sign but not sure if its same as that of a & b
2> tells a,x (and b) are of same sign but we don't know sign of y
combining => a,b,x,y are of same sign hence (-a,b), (-b,a), (-x, y) will be in the same quadrant.
infact (-a,x) (-b,x) (-a,y) (-b,y) will also be in the same quadrant.
This is sufficient. (even though we don't know which quadrant they will be in, may be in either 1st or 3rd quadrant depending on the sign)
This condition will not be called sufficient.
Originally Posted by MikeJung
Sign of x & y is independent of sign of a & b. Lets say we know that a & b are +ve . Can we say anything about x&y? No. x & y may still be +ve or -ve.
No clear answer means insufficient condition.
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