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- Thread starter Jamie
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- Jan 29, 2012

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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?

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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

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- Mar 5, 2012

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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?

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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)

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- Mar 5, 2012

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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.

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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.

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- #8

- Mar 5, 2012

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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?

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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?

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- #10

- Mar 5, 2012

- 9,595

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}