### Welcome to our community

#### Jamie

##### New member
The question is to classify/describe the following degenerate quadratic surface:

x2 - 2xy +2y2 - 2yz + z2 = 0

#### HallsofIvy

##### Well-known member
MHB Math Helper
The question is to classify/describe the following degenerate quadratic surface:

x2 - 2xy +2y2 - 2yz + z2 = 0
Write it as $$(x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0$$

Does that give you any ideas?

#### Jamie

##### New member
Write it as $$(x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0$$

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
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
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
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
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
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
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
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
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).