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Is point A closer to point (1,2) than to point (2,1)? 1.

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Is point A closer to point (1,2) than to point (2,1)? 1. [#permalink] New post 15 Dec 2007, 13:48
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Is point A closer to point (1,2) than to point (2,1)?

1. Point A lies on the line y=x
2. Point A lies on the line y=-x


Is there an algebraic/formula approach to this
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 [#permalink] New post 15 Dec 2007, 14:22
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(incorrect. Sorry guys, I misread the question. Perhaps, I'm too sleepy :shock: )
D

The geometrical approach is easer and faster for me.
a pure algebraic approach seems to be too unwieldy.

some simpler algebraic approach.

1. y=x.

we try to shift line to the a new place that a new line goes through our points.

point (1,2): y=x+1, shift is equal 1 to right
point (2,1): y=x-1, shift is equal 1 to left
so, points are equidistant.
SUFF.

2. y=-x

we try to make the same procedure with shifts.

point (1,2): y=-x+3, shift is equal 3 to right
point (2,1): y=-x+3, shift is equal 3 to right
so, points are equidistant.
SUFF.

Last edited by walker on 15 Dec 2007, 14:30, edited 1 time in total.
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answer [#permalink] New post 15 Dec 2007, 14:24
Answer A is right.

The best way to approach this is to draw the points and the lines in a cartesian plane.
With condition 1, each point on the line is equally distant from both the points, therefore it is SUFFICIENT to answer NO

With condition 2, it is simplt to visualize the some points on the line are closer to (1,2) and other points are closer to (2,1)
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Re: distance from a line [#permalink] New post 29 May 2009, 08:39
bmwhype2 wrote:
Is point A closer to point (1,2) than to point (2,1)?

1. Point A lies on the line y=x
2. Point A lies on the line y=-x

Is there an algebraic/formula approach to this


A should be it.

1. If point A lies on the line y=x, any point on this line has equal value for x and y and both are either positive or negative. Lets say point A lies on (4, 4).

i. The distance between A and (1, 2 ) = Sqrt {(4-1)^2 + (4-2)^2} = Sqrt {9 + 4 } = Sqrt (13)
ii. The distance between A and (2, 1 ) = Sqrt {(4-2)^2 + (4-1)^2} = Sqrt {4 + 9} = Sqrt (13)
Both are same.

If point A lies on (-3, -3).

i. The distance between A and (1, 2 ) = Sqrt {(-3-1)^2 + (-3-2)^2} = Sqrt (41)
ii. The distance between A and (2, 1 ) = Sqrt {(-3-2)^2 + (-3-1)^2} = Sqrt (41)

Both are equidistance. Suff..

2. If point A lies on the line y=-x, any point other than (0, 0) on this line has different values for x and y i.e. if x is +ve, y is -ve and vice versa. So any point on the line y = -x is not equidistance from point (1,2) and (2,1).

Lets say point A lies on (4, -4).

i. The distance between A and (1, 2 ) = Sqrt {(4-1)^2 + (-4-2)^2} = Sqrt {9 + 36 } = Sqrt (45)
ii. The distance between A and (2, 1 ) = Sqrt {(4-2)^2 + (-4-1)^2} = Sqrt {4 + 25} = Sqrt (29)

So these two points are not equidistance from A. NSF. Hence only A is correct.
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Re: distance from a line [#permalink] New post 30 May 2009, 07:10
Clearly A.

Totally agree, it is hard to look up even simple tricks when you sleepy.
Re: distance from a line   [#permalink] 30 May 2009, 07:10
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