MHB What is the classification of this degenerate quadratic surface?

[Jamie2]

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

x² - 2xy +2y² - 2yz + z² = 0

 

[HallsofIvy]

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?

 

[Jamie2]

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

 

[I like Serena]

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?

 

[Jamie2]

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

 

[I like Serena]

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.

 

[Jamie2]

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?

 

[I like Serena]

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?

 

[Jamie2]

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?

 

[I like Serena]

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]

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).