Question

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|

Answer 1

The quadrant in which the point (x, y) lies is Quadrant I, and the condition that determines this is -x < -y < |y|.

Let's analyze each condition separately:

(1) |xy| + x|y| + |x|y + xy > 0

This condition can be simplified using the triangle inequality:

|xy| + x|y| + |x|y + xy ≥ 0

This inequality holds true for all values of x and y, regardless of their signs. Therefore, this condition does not provide any information about the quadrant in which the point (x, y) lies.

(2) -x < -y < |y|

This condition can be rewritten as:

x > y > 0

This condition indicates that the point (x, y) lies in Quadrant I, where both x and y are positive.