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2. Let € be the conicx? +y2 _ 9) Determine the affine type Of € in the embedding plane z~y = 1, b) Find T(€) for the collineation T and determine ...

Question

2. Let € be the conicx? +y2 _ 9) Determine the affine type Of € in the embedding plane z~y = 1, b) Find T(€) for the collineation T and determine its affine type in theembedding plane z = 1.

2. Let € be the conicx? +y2 _ 9) Determine the affine type Of € in the embedding plane z~y = 1, b) Find T(€) for the collineation T and determine its affine type in the embedding plane z = 1.



Answers

$19-26$ Find a parametric representation for the surface.
\begin{equation}
\begin{array}{l}{\text { The part of the ellipsoid } x^{2}+2 y^{2}+3 z^{2}=1 \text { that lies to the }} \\ {\text { left of the } x z \text { -plane }}\end{array}
\end{equation}

For this problem, we want to find a parametric representation for the part of the plane Z equals X plus three that lies inside of the cylinder, X squared plus y squared equals one. Now there are a few different ways that we can go about this, but the easiest for us is going to be essentially using sort of polar coordinates for figuring out two X and Y. So we know that when translating between polar coordinates and Cartesian coordinates we have that X is going to equal are times coasts of data. Why is going to equal our time sign of data? And then there is the catch in polar or cylindrical coordinates. I should be saying that Zed is not necessarily specified ahead of time here, but for us we are told how's it relates to X. So I have that said is going to equal arcos data plus three. The last thing that we have is that we want part of that plane that lies inside of the cylinder, X squared plus y squared equals one. Well, what we can do here is recognize, we essentially have our plane, we have our cylinder that's you know, going up sort of through it with 00 in the center. So essentially what we have here is are is going to be able to vary, starting at zero. You know, we want to be able to get that 00 point there and then our should be able to vary so that we can get okay at that point and that point and that point close to the edge and so on what that lies inside of the X cylinder X squared plus y squared equals one requirement gives us is boundaries on our specifically we can define our surface as I've written here with the requirement that zero is less than the absolute value of our is less than are actually supposed zero is less than or equal to the absolute value of our has to be less than one, though less than or equal to might be an acceptable answer. That's going to be a little bit more subjective in the interpretation of the question, but less than is the safe assumption here and then we can just say that zero is going to be less than or equal to theta, which is going to be less than to buy. Yeah.

For this problem, we want to find a parametric representation for the part of the cylinder. X squared plus X squared equals nine that lies above the xy plane. And between the planes Y equals negative four and Y equals four. So the first thing that we can do here is recognize that were were asked about the cylinder. Uh, we're not told anything about what why has to be So why is going to be a parameter? Can say that? Why is going to equal you? And we just know that we need to have negative four. Be less than or equal to you, be less than or equal to be or not in the excuse me, positive for the next thing that we can do here is recognized that given the equation for the cylinder, we can rather easily rearrange that to get either X. In terms of Zed or Z. In terms of X. And then parameter is whatever we have in terms of. So let's get X. In terms of zed, you can subtract that squared from both sides. We get x squared equals nine minus x squared, which then means that x is going to equal plus or minus the square root of nine minus said square. Now comes for the significance about the lies above the xy plane for it to lie above the xy plane. We need to take the positive route here. Then we have our parametric representation. Our view V is going to equal. That is going to be the square root of nine minus V squared. We're parameter rising zed to equal V here, I hat plus. Then we have just you J hat plus. Then we have our zed or our V K hat.

So left off the have super means. Why has the last word With zero we can choose to variable, you know, freeway. So actually who's using was beat and we solve y for the Parametric representation of y the one minus you square by three fi square over to take a square root because we want y to be that non positives. So we put a negative sign here.

Were given a surface and were asked to premature eyes. This surface, the surface is a tilted plane inside a cylinder, specifically the portion of the plain X plus y plus Z equals one. And in part A This is going to be inside the cylinder X squared plus y squared equals nine. So because we are describing a surface which lies inside a cylinder, it might be a good idea to use cylindrical coordinates. Now, this cylinder X squared plus y squared equals nine. This lies wrapped around the Z axis Tell we'll take extra b r cosine data and why to be our sign data. Now we have Z is equal to one minus X minus y, which is, of course, equal to one minus are cosine data plus sign of theater. And so the parameter ation privatization for the surface is our of our theta equals R CoSine data I plus our sign data J plus one minus. R. CoSine data minus are signed data. Okay, Now there are no restrictions on theta. So theta can lie anywhere between zero and two pi mhm and find restrictions in our recall that are cylinder in part A. This has a radius of nine. So are is going to lie between zero and three. Sorry, Has a radius of three not raised? Not. And when Part B once again, we have the portion of the plain X plus y plus Z equals one that's inside a new cylinder. Why squared Plus C squared equals nine. This is now a cylinder that wraps around the X axis with the radius of three. So once again, we'll use cylindrical coordinates. We'll have why equals r cosign data Z equals R sine of fada and notice that X is equal to one minus Y minus Z, which is one minus R CoSine theta minus are signed data. And so putting this all together the parameter ization our of our theta is going to be one minus R CoSine theta minus are signed data I plus R CoSine data J plus our signed data. Okay. And we see that there is no restriction on status aesthetic and lie anywhere from 0 to 2 pi. Oh, and recall that the radius of the cylinder was three so that our lies between zero and three


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