Problem 53 Medium Difficulty

Graph the surfaces $ z = x^2 + y^2 $ and $ z = 1 - y^2 $ on a common screen using the domain $ \mid x \mid \le 1.2 , \mid y \mid \le 1.2 $ and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the $ xy $-plane is an ellipse.

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WZ

Wen Z.

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Mhm. Yes, I think uh So what we want to do is it's good stuff. These are process it's Pendergraph G equal to that's the tablets like square and She got 1 -9 sq on a common screen Using and we want to come and the moment for us is absolute value picks is less than equal to 1.2. International number of varieties, these other domains. So what it means is that uh for us, x is really going to lie From 1.2 to -1.2. Similarly why is not a -1.2 for one point. Okay. So first let us draw the grabs. Okay? Yeah. So let's say this is my G access and this is my excitement, right? And this is my so no you can see that as the values of G. Right? And G cannot be negative because he is always positive. So I'll start with equal to zero, which means that point X and Y zero. Then I go to G equal to one. Let's say then what it means is that if G equals to all, then we get access for for us by historical the one who is a something This minister at equaled one. I get a circle. The circle is like this. The radius is one here Similarly equal to two. We get a circle of radius squared of two. So the radius length increases. What it means is that this is really something like this. Yeah. So really it is this cylinder that you join this? It becomes a cylinder, right decision that this opens to the trusted G. Thing. This is what we call texas purpose widespread looks like And then I have equal to one -Y school. Now if you recall this is safe equal to one minus y squared that we right here. So what you get is that why square equal to minus of g minus mint. So this is really a parabola on the plan and it is opening downwards. So how does it look like on the Y Z plane? I have a parabola. So this means that is going to right so it is going to look something like this and the white deeper imagining that this is in the white plains. Of course it's not going to pass to the origin. So what does it pass through G equal to one? We get Y 20 So it passes to the G. Access on the .1. You said this is 0.01. And then when G called zero we get Y two K plus or minus one. So this means that to do something like this, this is how it looks, was part of 0 1.0 At this point is 0 -1. This is how the graph today. Yeah. Oh okay. And then the next question is too so that if we present it onto the white to the X. Y plane, we should get an ellipse. So let us project onto the xy plane. So what does projection that protects yplan mean? Yeah. So projection onto the xy plane really means is that I plug in G two B zero. So look at the intersection. Look at the intersection. Mhm. Of these graphs and then you plug in. So let us look at the intersection to intersection. Really means that you're intersex actually means that I have G. These two guys must be equal to each other. So I showed them equal to each other. Success square plus Y squared must be equal to one minus y squared. Remember certain to be zero and looking at the intersection. Mhm. True. Yeah. So this means that extra square plus two Y squared called one. So this means that excess square upon one Plus y square upon to over one. This must be one. This is wonderful. And hello? So this is an ellipse what you can and yeah.