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Say our question requires us to use traces to sketch the quadric surface with equation $x^2+y^2/9+z^2/4=1$. This is a worked example in our book, but I'm confused by their steps. So first we set $z=k$ to get $x^2+y^2/9=1-k^2/4$. First of all, is $k$ just a constant? And what is the difference between setting $z=k$ rather than $z=0$? Also, if $k$ is a constant (which I'm not sure if it is or not, or what it even represents for that matter), then how come when graphing using Wolfram, if I put in the equation with $k$, I get an ellipsoid, but if I just substitute say $0.5$ for $k^2/4$, I get a circle? And so then we repeat the same steps above with $x$ and $y$, and the same questions apply.

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Yes, $k$ is a constant. The way you should think of this is that $z=k$ is the plane parallel to the $xy$-coordinate plane at "height" $k$. So $k=0$ corresponds to exactly the $xy$-plane, and as $k$ increases you are getting different planes higher above that one and $k<0$ gives you planes "below" the $xy$-coordinate plane.

Substituting in different numbers gives you a "slice" of the 3D picture, so it is just the curve that you get at that height. This is why when you substitute in a value for $k$ you get just a curve. When you typed it into Wolfram with the $k$, it interpreted $k$ as a variable instead of a constant so it gave you the full 3D ellipsoid. When you just look at a plane section of the ellipsoid you get ellipses (or circles).

The purpose of this method is that if you know what a bunch of the 2D slices of the 3D thing look like (which is much easier since you just keep getting curves that should be familiar), then you can piece them together to sketch the 3D thing.

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And so when I'm graphing this by hand and for the first step where I set z=k, do I just plot that as a one dimensional ellipse? How do I plot that part? How do I treat the $k^2/4$? –  maq Feb 10 '11 at 3:13
    
I'm not sure what your book does for this, but the main point is to notice that below $z=-2$, you get empty. Nothing is below that. At $z=-2$ you get a point (0,0,2). Then as $k$ goes from $-2$ to $0$, the size of the ellipses gets bigger. Then from $0$ to $2$ they get smaller back to a single point. Then higher up than $2$, is empty again. Maybe on the same 2D, $xy$ graph you could put in $k=-2$, $k=-1$, $k=0$, $k=1$, and $k=2$ and label them like that. –  Matt Feb 10 '11 at 3:21

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