- Thread starter
- #1

- Thread starter Jamie
- Start date

- Thread starter
- #1

- Jan 29, 2012

- 1,151

Write it as [tex](x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0[/tex]The question is to classify/describe the following degenerate quadratic surface:

x^{2}- 2xy +2y^{2}- 2yz + z^{2}= 0

Does that give you any ideas?

- Thread starter
- #3

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)

but I don't know how to use that to help me describe the quadratic surface

- Admin
- #4

- Mar 5, 2012

- 8,774

Hi Jamie! Welcome to MHB!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

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?

- Thread starter
- #5

That they are equal to each other?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?

Or that (x-y)

- Admin
- #6

- Mar 5, 2012

- 8,774

That they are both zero!That they are equal to each other?

Or that (x-y)^{2}= -(y-z)^{2}

If either of them would be not zero, the sum would be positive, and therefore not equal to 0.

- Thread starter
- #7

right, I knew that too. But what does that mean for the equation's 3-dimensional surface?That they are both zero!

If either of them would be not zero, the sum would be positive, and therefore not equal to 0.

- Admin
- #8

- Mar 5, 2012

- 8,774

It means that $x=y$ and $y=z$.right, I knew that too. But what does that mean for the equation's 3-dimensional surface?

Both are equations of planes.

The

Where do they intersect?

- Thread starter
- #9

on the y axis? is the degenerate surface just a line?It means that $x=y$ and $y=z$.

Both are equations of planes.

Thedegeneratedquadratic surface is where they intersect.

Where do they intersect?

- Admin
- #10

- Mar 5, 2012

- 8,774

Indeed, the degenerate surface is just a line... but it is not the y axis...on the y axis? is the degenerate surface just a line?

Try to find a point that is on the line...

- Jan 29, 2012

- 1,151

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 theThat they are equal to each other?

Or that (x-y)^{2}= -(y-z)^{2}