akshatgupta87 wrote:Q.)In which quadrant of the coordinate plane does the point (x, y) lie?
(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y
Someone explain...
1) |xy| + x|y| + |x|y + xy > 0;
case 1) consider x>0; y>0; we have
xy+xy+xy+xy>0 4xy>0; which is true as both x and y are positive;
case 2) consider x>0; y<0; we have
-xy-xy+xy+xy>0;
0>0 which is false;
case 3) x<0; y>0;
-xy+xy-xy+xy>0;
0>0; which is false;
case 4) x<0; y<0;
xy-xy-xy+xy>0;
0>0; which is false;
hence out of all possible case only 1 case holds true, i.e. x>0; and y>0; hence x and y lies in the first quadrant. therefore 1 is sufficient.
2) -x<-y<|y|;
-y<|y|;
if y>0;
-y<y; which is true;
if y<0
-y<-y which is false; hence y is positive;
now consider -x<-y; if x becomes negative then -(-x) becomes positive, therefore our inequality -x<-y won't hold true, because positive quantity is always greater than negative;
therefore x should be positive;
as x and y should be positive therefore it must lie in the first quadrant, hence 2 alone is also sufficient to answer the question..!!!
hence
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