Absolute value + Geometry

[apoorva.srivastva]

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|

please explain st 1.

OA is
D

[4GMAT_Mumbai]

Very interesting sum. Thanks Apoorva.

Statment 1:

If x < 0 and y < 0; then the LHS of the expression will be 0. Taking simple examples works. let x = -2 and y = -3.
If x < 0 and y > 0; then the LHS of the expression will be 0. let x = -2 and y = 3.
If x > 0 and y < 0; then the LHS of the expression will be 0. let x = 2 and y = -3.
[spoiler]Only IF x > 0 and y > 0; will the LHS be > 0. Hence, (x,y) lies in Quadrant 1.[/spoiler]

Statement 2:

-y < |y| implies that y is a +ve number.

If -x < a negative number; then x has to be a +ve number. [spoiler]Thus, (x,y) lies in Q1. [/spoiler]

Hence D

Hope this helps.

[ajith]

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|

please explain st 1.

OA is
D

1) there are 4 cases a) x and y positive b) x and y negative c) x positive and y negative d) x negative and y positive

In each of the cases a,b,c and d (x,y) will lie on a different quadrant

in the case a

1) is satisfied

In the case b) c) and d)

1) is not satisfied since the sum is ZERO

so 1) is good enough to conclude that the point is in the first quadrant

2) it is apparent that y is +ve and since -x< -y => x>y x also should be +ve
Sufficient too

D

[sk818020]

The way I thought about (1) is:

You can rewrite |xy| + x|y| + |x|y + xy as:

( lxl + x ) ( lyl + y)

Which tells you:

If x is negative, then lxl + x = 0. If x positive, lxl + x = 2x.

and

If y is negative, then lyl + y = 0. If y is postive, lyl + y = 2y.

Thus, if ( lxl + x ) ( lyl + y) > 0, then neither y nor x can be negative because if they were ( lxl + x ) ( lyl + y) = 0.

Further, if x nor y is negative, then (x,y) must be in quadrant I (or top right quadrant) because they are both positive.

Hope that helps.