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Intern  G
Joined: 21 May 2016
Posts: 27
If x and y are non zero integers and a is a positive integer, is |x| =  [#permalink]

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4 00:00

Difficulty:   65% (hard)

Question Stats: 57% (02:51) correct 43% (02:50) wrong based on 48 sessions

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If x and y are non zero integers and a is a positive integer, is |x| = |y|?

(1) |x - y| = 2|x - a|

(2) |y - a| = |y|

Originally posted by a70 on 11 Aug 2018, 13:24.
Last edited by a70 on 14 Aug 2018, 04:14, edited 1 time in total.
Math Expert V
Joined: 02 Aug 2009
Posts: 8290
Re: If x and y are non zero integers and a is a positive integer, is |x| =  [#permalink]

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a70 wrote:
If x and y are non zero integers and a is a positive integer, is |x| = |y|?

(1) |x - y| = 2|x - a|

(2) |y - a| = |y|

So it means - is the numeric value of X and y same?

1) |x-y|=2|x-a|
You can easily see that it is possible both ways..
Say $$|x|\neq{|y|}$$
X=4, y=2 so x-y=4-2=2...... therefore 2|x-a|=2 or x=a+1.....a=3... possible
Say x=y
So 2x=2|x-a|.........x=|x-a|......a=2x......a can take positive value.. possible
Insufficient

2) |y-a|=|y|
Square both sides
$$y^2-2ay+a^2=y^2......a(a-2y)=0$$
So either a=0 or a=2y
a is positive so a=2y
Insufficient

Combined
|x-y|=2|x-2y|.......
Square two sides
$$x^2-2xy+y^2=4x^2-16xy+16y^2............3x^2-14xy+15y^2=0$$

Since |x|=|y|......
x^2=y^2....... So 3x^2+15y^2=18x^2 and
14xy will be 14x^2 if x=y or -14x^2 if x=-y

Thus 3x^2+15y^2=18x^2, which will not be equal to 14x^2 or -14x^2
Hence Ans is NO
Sufficient

C
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SVP  V
Joined: 26 Mar 2013
Posts: 2343
Re: If x and y are non zero integers and a is a positive integer, is |x| =  [#permalink]

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chetan2u wrote:
a70 wrote:
If x and y are non zero integers and a is a positive integer, is |x| = |y|?

(1) |x - y| = 2|x - a|

(2) |y - a| = |y|

So it means - is the numeric value of X and y same?

1) |x-y|=2|x-a|
You can easily see that it is possible both ways..
Say $$|x|\neq{|y|}$$
X=4, y=2 so x-y=4-2=2...... therefore 2|x-a|=2 or x=a+1.....a=3... possible
Say x=y
So 2x=2|x-a|.........x=|x-a|......a=2x......a can take positive value.. possible
Insufficient

2) |y-a|=|y|
Square both sides
$$y^2-2ay+a^2=y^2......a(a-2y)=0$$
So either a=0 or a=2y
a is positive so a=2y
Insufficient

Combined
|x-y|=2|x-2y|.......
Square two sides
$$x^2-2xy+y^2=4x^2-16xy+8y^2............3x^2-14xy+7y^2=0$$
Since |X|=|y|......x^2=y^2 and 14xy will be 14x^2 or -14x^2
3x^2+7y^2=10x^2, which will not be equal to 14x^2 or -14x^2
Hence Ans is NO
Sufficient

C

Hi

I think there is a small mistake here: it must be

x^2-2xy+y^2=4 (x^2-4xy+4y^2)=............3x^2-14xy+15y^2=0
Math Expert V
Joined: 02 Aug 2009
Posts: 8290
Re: If x and y are non zero integers and a is a positive integer, is |x| =  [#permalink]

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Mo2men wrote:
chetan2u wrote:
a70 wrote:
If x and y are non zero integers and a is a positive integer, is |x| = |y|?

(1) |x - y| = 2|x - a|

(2) |y - a| = |y|

So it means - is the numeric value of X and y same?

1) |x-y|=2|x-a|
You can easily see that it is possible both ways..
Say $$|x|\neq{|y|}$$
X=4, y=2 so x-y=4-2=2...... therefore 2|x-a|=2 or x=a+1.....a=3... possible
Say x=y
So 2x=2|x-a|.........x=|x-a|......a=2x......a can take positive value.. possible
Insufficient

2) |y-a|=|y|
Square both sides
$$y^2-2ay+a^2=y^2......a(a-2y)=0$$
So either a=0 or a=2y
a is positive so a=2y
Insufficient

Combined
|x-y|=2|x-2y|.......
Square two sides
$$x^2-2xy+y^2=4x^2-16xy+8y^2............3x^2-14xy+7y^2=0$$
Since |X|=|y|......x^2=y^2 and 14xy will be 14x^2 or -14x^2
3x^2+7y^2=10x^2, which will not be equal to 14x^2 or -14x^2
Hence Ans is NO
Sufficient

C

Hi

I think there is a small mistake here: it must be

x^2-2xy+y^2=4 (x^2-4xy+4y^2)=............3x^2-14xy+15y^2=0

Thanks a lot...
_________________ Re: If x and y are non zero integers and a is a positive integer, is |x| =   [#permalink] 12 Aug 2018, 05:01
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