RBBmba@2014 wrote: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|
Statement 1: |xy| + x|y| + |x|y + xy > 0
Here, it is clear that xy≠0.
Test EASY CASES in |xy| + x|y| + |x|y + xy.
Case 1: x=1, y=1 --> |1*1| + 1|1| + |1|1 + 1*1 = 4.
Case 2: x=-1, y=1 --> |(-1)(1)| + -1|1| + |-1|1 + -1*1 = 0.
Case 3: x=1, y=-1 --> |1*-1| + 1|-1| + |1|-1 + (1)(-1) = 0.
Case 4: x=-1, y=-1 --> |(-1)(-1)| + (-1)|-1| + |-1|(-1) + (-1)(-1) = 0.
Only Case 1 satisfies the constraint that |xy| + x|y| + |x|y + xy > 0.
Thus, (x,y) must be (+, +), with the result that it lies in Quadrant I.
SUFFICIENT.
Statement 2: -x < -y < |y|
Here, it is clear that y≠0, implying that |y| > 0.
Thus:
-x < -y < 0 < |y|
Multiplying by -1 and flipping the inequality symbols, we get;
x > y > 0 > -|y|.
Since x > y > 0, (x,y) must lie in Quadrant I.
SUFFICIENT.
The correct answer is
D.
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