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Degenerate Quadratic Surface

Jamie

New member
Feb 11, 2014
17
The question is to classify/describe the following degenerate quadratic surface:

x2 - 2xy +2y2 - 2yz + z2 = 0
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
The question is to classify/describe the following degenerate quadratic surface:

x2 - 2xy +2y2 - 2yz + z2 = 0
Write it as [tex](x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0[/tex]

Does that give you any ideas?
 

Jamie

New member
Feb 11, 2014
17
Write it as [tex](x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0[/tex]

Does that give you any ideas?

well that's the same as (x-y)2 + (y-z)2 = 0
but I don't know how to use that to help me describe the quadratic surface
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
well that's the same as (x-y)2 + (y-z)2 = 0
but I don't know how to use that to help me describe the quadratic surface
Hi Jamie! Welcome to MHB! :)

Did you know that a square is always at least zero?
Suppose the sum of 2 squares is equal to zero, what does that say about those squares?
 

Jamie

New member
Feb 11, 2014
17
Hi Jamie! Welcome to MHB! :)

Did you know that a square is always at least zero?
Suppose the sum of 2 squares is equal to zero, what does that say about those squares?
That they are equal to each other?
Or that (x-y)2 = -(y-z)2
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
That they are equal to each other?
Or that (x-y)2 = -(y-z)2
That they are both zero!
If either of them would be not zero, the sum would be positive, and therefore not equal to 0.
 

Jamie

New member
Feb 11, 2014
17
That they are both zero!
If either of them would be not zero, the sum would be positive, and therefore not equal to 0.
right, I knew that too. But what does that mean for the equation's 3-dimensional surface?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
right, I knew that too. But what does that mean for the equation's 3-dimensional surface?
It means that $x=y$ and $y=z$.
Both are equations of planes.
The degenerated quadratic surface is where they intersect.
Where do they intersect?
 

Jamie

New member
Feb 11, 2014
17
It means that $x=y$ and $y=z$.
Both are equations of planes.
The degenerated quadratic surface is where they intersect.
Where do they intersect?
on the y axis? is the degenerate surface just a line?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
on the y axis? is the degenerate surface just a line?
Indeed, the degenerate surface is just a line... but it is not the y axis...
Try to find a point that is on the line...
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
That they are equal to each other?
Or that (x-y)2 = -(y-z)2
Both! The only way a sum of squares can be 0 is if each is 0. x- y= 0 and y- z= 0 which is the same as the z= y= x. That is the line through (0, 0, 0) and (1, 1, 1).