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# Which of the following points is closest to line y=x?

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Which of the following points is closest to line y=x? [#permalink]

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21 Nov 2007, 07:14
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Which of the following points is closest to line y=x?

A. (2, -1)
B. (2, 4)
C. (-1, 1)
D. (2, 1)
E. (-2, 0)

M12-20
[Reveal] Spoiler: OA

Last edited by Bunuel on 01 Jul 2012, 06:55, edited 1 time in total.
Edited the question and added the OA.
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21 Nov 2007, 08:20
looks like (2,1)...just closest to 1:1 ratio...whats the OA?
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Re: PS closest point [#permalink]

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21 Nov 2007, 20:24
bmwhype2 wrote:
Which of the following points is closest to line y=x?

(2, -1)
(2, 4)
(-1, 1)
(2, 1)
(-2, 0)

How can i solve this with a mathematical approach rather than drawing it out?

Well I am still struggling to find a gmat solution ( nice and easy ) for this. Kept waiting that some one will solve.

Here's my take

Drop a perpendicular from the point to the line y=x. Slope will be -1 for all the perps which are dropped on this line

The point where it hits will be have same x and y coordinates. Assume its a
So a - (-1) / a - 2 = -1 => a + 1 = 2 -a => a = 1 .
Now we can calculate the distance by
sqrt [ ( a - 2 ) ^ 2 + ( a + 1 ) ^2 ] ...no need to take square root , the smallest of the values .
Again few points can be ignored by just viewing on a rough sketch.
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Re: PS closest point [#permalink]

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22 Nov 2007, 00:02
bmwhype2 wrote:
Which of the following points is closest to line y=x?

(2, -1)
(2, 4)
(-1, 1)
(2, 1)
(-2, 0)

How can i solve this with a mathematical approach rather than drawing it out?

D. can be solved by finding the length between each answer option and x=y point.

Also can be found drawing a rough sketch and pointing the points in answer choices.
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Re: PS closest point [#permalink]

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22 Nov 2007, 00:25
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bmwhype2 wrote:
Which of the following points is closest to line y=x?

(2, -1)
(2, 4)
(-1, 1)
(2, 1)
(-2, 0)

How can i solve this with a mathematical approach rather than drawing it out?

Here is a formula which can be use to canculate the distance between a point (m,n) and a general line L : aX+bY+C=0

d=abs(am+bn+c)/sqrt(a^2+b^2)

Take this question for instance :
L x-y=0 ---> a=1, b=-1
then you can get the distance for the first choice 2*1+(-1)*(-1)+0/sqrt[1^2+(-1)^2)]
Actually for this question you dont need nessisarily to work the distance out to get the smallest one
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Re: Which of the following points is closest to line y=x? (2, [#permalink]

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01 Jul 2012, 06:50
can someone please explain me how you guys solved this question using graphical approach? If someone can post the actual graphic solution, that will be helpful.
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Re: Which of the following points is closest to line y=x? (2, [#permalink]

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01 Jul 2012, 06:58
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Expert's post
teal wrote:
can someone please explain me how you guys solved this question using graphical approach? If someone can post the actual graphic solution, that will be helpful.

Which of the following points is closest to line y=x?

A. (2, -1)
B. (2, 4)
C. (-1, 1)
D. (2, 1)
E. (-2, 0)

Look at the diagram below:
Attachment:

m12-20.png [ 12.86 KiB | Viewed 4753 times ]
As you can see point (2, 1) is the closest to line y=x.

Answer: D.

Hope it helps.
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Re: Which of the following points is closest to line y=x? [#permalink]

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01 Jul 2012, 07:45
Thanks a lot for getting back. Distance of points from the line (that we need to consider) is the length of the perpendicular dropped on the line from each point? Please confirm. Also, how to get such a good image (with an almost accurate estimate) on the exam scratch pad?
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Re: Which of the following points is closest to line y=x? [#permalink]

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04 Jul 2012, 02:41
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I solved like this :

y=x line is given
=> x-y = 0

now for all the option choice do X-Y operation
1. 2-(-1)=3
2. 2 -4 =-2
3. -1-1=-2
4. 2-1 = 1
5. -2-0 = -2

Clearly option 4 is nearest to 0 which is our equation x=y...

Am I right on this approach?
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Re: Which of the following points is closest to line y=x? [#permalink]

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05 Jul 2012, 19:55
Bunnel, please confirm if Bhavin's approach will always work?

Visual approach worked out fine this time for me, I drew the figure on my pad just like the one Bunnel has drwan......I utilized the square grid of my pad for drawing the graph, it worked out...now I am glad I will not lose this question. Thanks, Bunnel.
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Re: Which of the following points is closest to line y=x? [#permalink]

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07 Aug 2012, 22:04
I think bunuel's approach is the right one for this. I do not know if you can substitute for x and y with the points given as per Bhavinshah's approach!
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Re: Which of the following points is closest to line y=x? (2, [#permalink]

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10 Sep 2012, 08:13
Bunuel wrote:
teal wrote:
can someone please explain me how you guys solved this question using graphical approach? If someone can post the actual graphic solution, that will be helpful.

Which of the following points is closest to line y=x?

A. (2, -1)
B. (2, 4)
C. (-1, 1)
D. (2, 1)
E. (-2, 0)

Look at the diagram below:
Attachment:
m12-20.png
As you can see point (2, 1) is the closest to line y=x.

Answer: D.

Hope it helps.

Hi bunuel,

Is there any other way to solve this problem apart from drawing perpendicular to the respective points or graphs

Waiting for your response
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Re: Which of the following points is closest to line y=x? (2, [#permalink]

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11 Sep 2012, 02:05
fameatop wrote:
Bunuel wrote:
teal wrote:
can someone please explain me how you guys solved this question using graphical approach? If someone can post the actual graphic solution, that will be helpful.

Which of the following points is closest to line y=x?

A. (2, -1)
B. (2, 4)
C. (-1, 1)
D. (2, 1)
E. (-2, 0)

Look at the diagram below:
Attachment:
m12-20.png
As you can see point (2, 1) is the closest to line y=x.

Answer: D.

Hope it helps.

Hi bunuel,

Is there any other way to solve this problem apart from drawing perpendicular to the respective points or graphs

Waiting for your response

From my point of view the best way to solve this question in to make a diagram as in my post above.
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Re: Which of the following points is closest to line y=x? [#permalink]

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05 Jan 2013, 00:33
bmwhype2 wrote:
Which of the following points is closest to line y=x?

A. (2, -1)
B. (2, 4)
C. (-1, 1)
D. (2, 1)
E. (-2, 0)

M12-20

My approach is to draw the points and then the x=y line.

(2,-1) is 3 units away to the right and to the bottom of the line
(2,4) is 2 units away to the top of the line
(-1,1) is 2 units to the left of the line
(2,1) is 1 unit to the bottom and to the right of the line
(-2,0) is 2 units to the right of the lne

Answer: D
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Re: Which of the following points is closest to line y=x? [#permalink]

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30 Apr 2013, 21:48
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If the line is ax+by+c =0 and Point ( x1, y1)

Distance = | ax1+ by1 + c | / sqrt( a^2 + b^2 )

Using this formulae one can easily plug in and find out the least distance
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Re: Which of the following points is closest to line y=x? [#permalink]

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28 Dec 2013, 07:24
shelrod007 wrote:
If the line is ax+by+c =0 and Point ( x1, y1)

Distance = | ax1+ by1 + c | / sqrt( a^2 + b^2 )

Using this formulae one can easily plug in and find out the least distance

I think it is much easier just to draw the points than to apply this formula under real GMAT conditions

Hope you consider it

PS. Is bhavinshah5685 approach valid anyone? Looks nice too

Cheers!
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Re: Which of the following points is closest to line y=x? [#permalink]

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20 Jul 2015, 07:29
Hi all, why do we consider only a line passing through Quadrants III and I, and not also a line passing through quadrants II and IV (X=-1, Y=-1 etc) ? In this case the answer would be C...
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Re: Which of the following points is closest to line y=x? [#permalink]

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20 Jul 2015, 07:33
BrainLab wrote:
Hi all, why do we consider only a line passing through Quadrants III and I, and not also a line passing through quadrants II and IV (X=-1, Y=-1 etc) ? In this case the answer would be C...

Have you checked this:

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Re: Which of the following points is closest to line y=x? [#permalink]

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20 Jul 2015, 07:39
BrainLab wrote:
Hi all, why do we consider only a line passing through Quadrants III and I, and not also a line passing through quadrants II and IV (X=-1, Y=-1 etc) ? In this case the answer would be C...

This is as per the question. It talks about y=x and this only passes through I and III quadrants. y=-x will be the line passing through II and IV quadrants.
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Which of the following points is closest to line y=x? [#permalink]

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20 Jul 2015, 07:43
Engr2012 wrote:
BrainLab wrote:
Hi all, why do we consider only a line passing through Quadrants III and I, and not also a line passing through quadrants II and IV (X=-1, Y=-1 etc) ? In this case the answer would be C...

This is as per the question. It talks about y=x and this only passes through I and III quadrants. y=-x will be the line passing through II and IV quadrants.

Thanks, was a silly mistake... we can not get values (-1,1) in th equiatoin y=x, was just to fast in drawing symmetrical fihures ))...
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Which of the following points is closest to line y=x?   [#permalink] 20 Jul 2015, 07:43
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# Which of the following points is closest to line y=x?

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