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

Question

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

M12-20

bhavinshah5685 · Accepted Answer

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?

Bunuel · Answer

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?

young_gun · Answer

looks like (2,1)...just closest to 1:1 ratio...whats the OA?

ashkrs · Answer

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   ...no need to take square root , the smallest of the values .
Again few points can be ignored by just viewing on a rough sketch.

Fistail · Answer

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.

Richard Lee · Answer

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  
Actually for this question you dont need nessisarily to work the distance out to get the smallest one

teal · Answer

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.

teal · Answer

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?

teal · Answer

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.

rajathpanta · Answer

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!

fameatop · Answer

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

Bunuel · Answer

From my point of view the best way to solve this question in to make a diagram as in my post above.

mbaiseasy · Answer

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

shelrod007 · Answer

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

jlgdr · Answer

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! J

BrainLab · Answer

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...

Bunuel · Answer

Have you checked this:
 https://gmatclub.com/forum/download/file.php?id=17724

ENGRTOMBA2018 · Answer

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.

BrainLab · Answer

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 ))...

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

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GMAT Problem Solving (PS) Questions

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

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