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Equation |x/2| + |y/2| = 5 encloses a certain region on the

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Equation |x/2| + |y/2| = 5 encloses a certain region on the [#permalink] New post 18 Aug 2008, 18:00
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Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?

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Please explain. Also how do you recognize when a particular equation refers to a square.
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Re: PS: Coordinate Geometry: Square [#permalink] New post 18 Aug 2008, 18:24
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x97agarwal wrote:
Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?

20
50
100
200
400

Please explain. Also how do you recognize when a particular equation refers to a square.


|x/2| + |y/2| = 5. So what is the x intercept ? put |y/2|=0. |x/2| = 5
so x/2 = 5 or x/2=-5 and hence x=10 or x=-10.

Similarly take |x/2| = 0. adn you get y=10 or -10. This will give you a square with vertices at (10,0) (0,10) (-10,0) (0,-10)

this is a square with side 10 * (sq root 2) so area = 200
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Re: PS: Coordinate Geometry: Square [#permalink] New post 18 Aug 2008, 18:33
Thanks Bhushan.

Good explanation.
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Re: PS: Coordinate Geometry: Square [#permalink] New post 18 Aug 2008, 21:20
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x97agarwal wrote:
Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?

20
50
100
200
400

Please explain. Also how do you recognize when a particular equation refers to a square.

Attachments

co-square.gif
co-square.gif [ 6.52 KiB | Viewed 1157 times ]


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Re: PS: Coordinate Geometry: Square [#permalink] New post 18 Aug 2008, 21:33
x97agarwal wrote:
Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?

20
50
100
200
400

Please explain. Also how do you recognize when a particular equation refers to a square.


seems the question is not complete.

it should be what is the "maximum" area of the square covered by the equation. then only it is 200. otherwise it could be anything 200 or lesst than 200.
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Re: PS: Coordinate Geometry: Square [#permalink] New post 18 Aug 2008, 23:04
gmattiger, I could not get why maximum area should be 200?
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Re: PS: Coordinate Geometry: Square [#permalink] New post 19 Aug 2008, 07:09
scthakur wrote:
gmattiger, I could not get why maximum area should be 200?


With the given equation, you can have many squares but only one with 200 area. You cannot have a square with the area > 200 with the given eq..
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Re: PS: Coordinate Geometry: Square [#permalink] New post 19 Aug 2008, 08:03
GMAT TIGER wrote:
scthakur wrote:
gmattiger, I could not get why maximum area should be 200?


With the given equation, you can have many squares but only one with 200 area. You cannot have a square with the area > 200 with the given eq..



With given 4 equations.... they intersect only at 4 points.. there only one square for the are enclosed by 4 lines.

Please note that given equation is |x/2| + |y/2| = 5 not |x/2| + |y/2| <= 5
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Re: PS: Coordinate Geometry: Square [#permalink] New post 20 Aug 2008, 07:46
x2suresh wrote:
GMAT TIGER wrote:
scthakur wrote:
gmattiger, I could not get why maximum area should be 200?


With the given equation, you can have many squares but only one with 200 area. You cannot have a square with the area > 200 with the given eq..



With given 4 equations.... they intersect only at 4 points.. there only one square for the are enclosed by 4 lines.

Please note that given equation is |x/2| + |y/2| = 5 not |x/2| + |y/2| <= 5


absolutely.
a region can be made with x = +/- 5 and y = +/- 5 and still yields |x/2| + |y/2| = 5, which doesnot mean that |x/2| + |y/2| <= 5.

there are other regions possible with the constraints but only x = y= +/- 10 gives the max area.

let me be even precise:

if x = 0 and y = +/- 10, |x/2| + |y/2| = 5
if x = +/- 10 and y = 0, |x/2| + |y/2| = 5
if x = +/-2 and y = +/-8, |x/2| + |y/2| = 5
if x = +/-4 and y = +/-6, |x/2| + |y/2| = 5
if x = +/-5 and y = +/-5, |x/2| + |y/2| = 5

you can have multiple values for x and y but the max area gives only when x = +/-10 and y = +/-10.
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Re: PS: Coordinate Geometry: Square [#permalink] New post 20 Aug 2008, 08:07
Gmat Tiger who are you? and how can I begin to think like you?
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Re: PS: Coordinate Geometry: Square [#permalink] New post 20 Aug 2008, 08:43
GMAT TIGER wrote:
[quote="x2suresh
With given 4 equations.... they intersect only at 4 points.. there only one square for the are enclosed by 4 lines.

Please note that given equation is |x/2| + |y/2| = 5 not |x/2| + |y/2| <= 5

absolutely.
a region can be made with x = +/- 5 and y = +/- 5 and still yields |x/2| + |y/2| = 5, which doesnot mean that |x/2| + |y/2| <= 5.

there are other regions possible with the constraints but only x = y= +/- 10 gives the max area.

let me be even precise:

if x = 0 and y = +/- 10, |x/2| + |y/2| = 5
if x = +/- 10 and y = 0, |x/2| + |y/2| = 5
if x = +/-2 and y = +/-8, |x/2| + |y/2| = 5
if x = +/-4 and y = +/-6, |x/2| + |y/2| = 5
if x = +/-5 and y = +/-5, |x/2| + |y/2| = 5

you can have multiple values for x and y but the max area gives only when x = +/-10 and y = +/-10.


Can you draw and show me in the picture.. with area less than 200 and enclosed by 4 lines
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Re: PS: Coordinate Geometry: Square [#permalink] New post 20 Aug 2008, 10:01
x2suresh wrote:
GMAT TIGER wrote:
[quote="x2suresh
With given 4 equations.... they intersect only at 4 points.. there only one square for the are enclosed by 4 lines.

Please note that given equation is |x/2| + |y/2| = 5 not |x/2| + |y/2| <= 5

absolutely.
a region can be made with x = +/- 5 and y = +/- 5 and still yields |x/2| + |y/2| = 5, which doesnot mean that |x/2| + |y/2| <= 5.

there are other regions possible with the constraints but only x = y= +/- 10 gives the max area.

let me be even precise:

if x = 0 and y = +/- 10, |x/2| + |y/2| = 5
if x = +/- 10 and y = 0, |x/2| + |y/2| = 5
if x = +/-2 and y = +/-8, |x/2| + |y/2| = 5
if x = +/-4 and y = +/-6, |x/2| + |y/2| = 5
if x = +/-5 and y = +/-5, |x/2| + |y/2| = 5

you can have multiple values for x and y but the max area gives only when x = +/-10 and y = +/-10.


Can you draw and show me in the picture.. with area less than 200 and enclosed by 4 lines


it is clear even without the drawing and lets say:

x = 5 and y = 5
x = 5 and y = -5
x = -5 and y = 5
x = -5 and y = -5

so the (x,y) co-ordinates of the four points are (5, 5), (5, -5), (-5, 5) and (-5, -5)

in this case, area is 100.
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Re: PS: Coordinate Geometry: Square [#permalink] New post 20 Aug 2008, 10:15
GMAT TIGER wrote:
x2suresh wrote:
GMAT TIGER wrote:
[quote="x2suresh
With given 4 equations.... they intersect only at 4 points.. there only one square for the are enclosed by 4 lines.

Please note that given equation is |x/2| + |y/2| = 5 not |x/2| + |y/2| <= 5

absolutely.
a region can be made with x = +/- 5 and y = +/- 5 and still yields |x/2| + |y/2| = 5, which doesnot mean that |x/2| + |y/2| <= 5.

there are other regions possible with the constraints but only x = y= +/- 10 gives the max area.

let me be even precise:

if x = 0 and y = +/- 10, |x/2| + |y/2| = 5
if x = +/- 10 and y = 0, |x/2| + |y/2| = 5
if x = +/-2 and y = +/-8, |x/2| + |y/2| = 5
if x = +/-4 and y = +/-6, |x/2| + |y/2| = 5
if x = +/-5 and y = +/-5, |x/2| + |y/2| = 5

you can have multiple values for x and y but the max area gives only when x = +/-10 and y = +/-10.


Can you draw and show me in the picture.. with area less than 200 and enclosed by 4 lines


it is clear even without the drawing and lets say:

x = 5 and y = 5
x = 5 and y = -5
x = -5 and y = 5
x = -5 and y = -5

so the (x,y) co-ordinates of the four points are (5, 5), (5, -5), (-5, 5) and (-5, -5)

in this case, area is 100.


But this will not cover the all area of this region by that equation. only part of it.see the picture.

Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?
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Re: PS: Coordinate Geometry: Square [#permalink] New post 20 Aug 2008, 10:51
x2suresh wrote:
GMAT TIGER wrote:
it is clear even without the drawing and lets say:

x = 5 and y = 5
x = 5 and y = -5
x = -5 and y = 5
x = -5 and y = -5

so the (x,y) co-ordinates of the four points are (5, 5), (5, -5), (-5, 5) and (-5, -5)

in this case, area is 100.


But this will not cover the all area of this region by that equation. only part of it.see the picture.

Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?


I agree with your statement highlighted above and that is also my point if the question doesnot say max area. Otherwise how do you find x,y co-ordinates? therefore, the question should say the max. area vcovered by the given equation. if the question doesnot say the max. area, then the equation could mean that (x,y) co-ordinates of the four points are (5, 5), (5, -5), (-5, 5) and (-5, -5).

Since many values for (x, y) satisfy the given equation, we cannot say the coordinate of (x,y) are (10, 0), (-10, 0), (0, 10) and (0, -10) .

The similar question also raised by "x97agarwal" as well.

correct me if any.


x97agarwal wrote:
Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?

20
50
100
200
400
Please explain. Also how do you recognize when a particular equation refers to a square.

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Re: PS: Coordinate Geometry: Square [#permalink] New post 20 Aug 2008, 12:00
GMAT TIGER wrote:
x2suresh wrote:
GMAT TIGER wrote:
it is clear even without the drawing and lets say:

x = 5 and y = 5
x = 5 and y = -5
x = -5 and y = 5
x = -5 and y = -5

so the (x,y) co-ordinates of the four points are (5, 5), (5, -5), (-5, 5) and (-5, -5)

in this case, area is 100.


But this will not cover the all area of this region by that equation. only part of it.see the picture.

Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?


I agree with your statement highlighted above and that is also my point if the question doesnot say max area. Otherwise how do you find x,y co-ordinates? therefore, the question should say the max. area vcovered by the given equation. if the question doesnot say the max. area, then the equation could mean that (x,y) co-ordinates of the four points are (5, 5), (5, -5), (-5, 5) and (-5, -5).

Since many values for (x, y) satisfy the given equation, we cannot say the coordinate of (x,y) are (10, 0), (-10, 0), (0, 10) and (0, -10) .

The similar question also raised by "x97agarwal" as well.

correct me if any.


x97agarwal wrote:
Equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane. What is the area of this region?

20
50
100
200
400
Please explain. Also how do you recognize when a particular equation refers to a square.


(10, 0), (-10, 0), (0, 10) and (0, -10) are not just ordinary co-ordinates they are points where two lines intersect.

Please answer to the following question.
What is the area of region enlcosed by the following equations.
x=5
x=-5
y=5
y=-5

a) 100
b) >100
c) <100
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Re: PS: Coordinate Geometry: Square [#permalink] New post 21 Aug 2008, 18:21
Can i make point here.

The points GMAT TIGER referring to (5,-5) (-5,5)(5,5)(-5,-5) satisfy the equations
|x| =5
|y|=5

in which case , all lines forming the square are parallels/perpendicular to x/y axis.
hence their slope are 0 or un-defined

They do not satisfy equation(s) |x|+|y|=10 since the slope of the equations is not undefined it is +/-1


So the area (dont even think about max or min) should be 200.

comments welcome
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Re: PS: Coordinate Geometry: Square [#permalink] New post 22 Aug 2008, 05:46
Expert's post
Tiger, the main point here is not geometry but something more common, I would say, something on default. Let's consider following examples:

- A boy has bought a box of candies in which there are 12 candies. How many candies has boy bought?
- a man has filled empty 20-gallons tanks of his car. How many litters has he filled?
- a cube has edges of 1 meters. What is volume enclosed by the cube?

In our daily life we would clearly say that answers are 12 candies, 20 gallons and 1 cubic meter. But in general we also can say that the boy has bought 3 candies, the man has filled 7 gallons and the volume is 0.35 cubic meters. Why do we say in other way? Because it is unambiguous and on default. :)

Hope this help
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Re: PS: Coordinate Geometry: Square   [#permalink] 22 Aug 2008, 05:46
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