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$11-20$ Use traces to sketch and identify the surface.$$9 x^{2}-y^{2}+z^{2}=0$$...

Question

$11-20$ Use traces to sketch and identify the surface.$$9 x^{2}-y^{2}+z^{2}=0$$

$11-20$ Use traces to sketch and identify the surface. $$9 x^{2}-y^{2}+z^{2}=0$$



Answers

Use traces to sketch and identify the surface.

$ x = y^2 + 4z^2 $

All right. They want us to use traces to figure out what this is a picture of. So first um let zb zero. So we're looking in the xy plane? So X squared equals X squared equals four Y squared. So x equals plus and -2 y. Or if it's easier for you to think about why equals plus minus two X. A half X. Sorry. Okay. So those are lines in the xy plane. Sex Y plane go through 00, slope is one half And -1 half. Okay. And then um in let Y equals zero. And then you get X squared equals Z squared. So X. Or Z equals plus minus X. So in the Z explain also lines but steeper. Here's excuse the these have slope one and -1. Oh okay. So then what about in the other plane you get uh let X equal zero. And so you get four Y squared plus Z squared equals zero. Okay so that's not happening Unless X&Y are both zero. Okay. Um So that's the zero. But how about if for example X. Is one then one squared equals four Y squared plus B squared. So that's any lips in the Y. Z plane um One squared equals y squared over 1/4. Was he squared? So it's like this. Alright let's put it all together. So for sure it goes to this point right here um And the in the X. What in the Wednesday was zero? So in the xy plane, its lines in the xz plane? Its lines. And in the Y Z. Plane it's ellipses. Okay, so in the which which way did the ellipses go? Um I got to look again fat. Tall on the z. Smaller on the line. Yeah. Okay. So here's here's a line in the XZ plane. Here's a line in the Xz plane. Here's a line in the UAE explain. Here's a line in the UAE explain. I kind of got a shifted a little bit. Okay, so it's a cone, it opens on the X. Axis on the positive X. Axis and it's not a circular comb but uh elliptical cone, that's um 1-1 half here or twice as big as twice as tall as it is wide. Okay, so here's how I got that. It was this that told me what to do. I got to have ellipses in the Y. Z. Plane. Okay. Button or parallel to the Y. Z. Plane? In the Y. Z. Plane. Just have a point. And behind the Y. Z. Plane, we don't have anything but coming out of the Y. Z. Plane, we have ellipses. All right,

In the problem. And we have given an equation that is did Squire minus four X Squire minus y square. That is equal to four. So here is that it's quite equal. Four plus four X Squire minus Y. Sorry this is plus Y square. So J equals route over off four mine plus four x square plus y squared. Now here we will put the value of X and y zero. Therefore zero is equal to No tomorrow. For that is equal to plus -2. Now The value of Jade equals plus -2. This is the real value of debt. Now here we have X equal so minus four X squared is equal to four minus jet square plus y squared. So here it is four get square and plus weisglass. Now ex Squire becomes minus one and it is Plus Jade Square upon four. My nurse, why square upon for our eggs becomes route ovtcharov minus one plus j disk wired upon food minus y squared upon before. Now when we put the value of a J D n Y zero. Therefore this equation becomes so X equals plus minus over time. So it's a complex number now. For value of why we have Y squared equal food Or rather it is -4 minus jade Squire Plus four x squared. So why is equal the route over half minus four minus that Squire plus four X square. So here kill those mistakes here. So we have Y equals plus minus two area. So this is also a complex number X and Y is complex number. But jade is a real number. Therefore the center of the curve lies on jet equals plus minus two. So this is the point, that is, this is the mm point. This is too, And this is also -2. We have ex excess Y axis and jx so J equals plus minus two is the value, and X and y are the complex numbers. So we have The value of debt equals -2. Sorry, there's a blast to hear and -2 here. So this is the car for this equation and this is the answer to the problem.

So what do you want to do is we want to draw the graph access square? Lost 36 equals to nine y square The last four G school. No, you observed one thing first start with the stress of case. So if X equals to stress effects in Mexico. Okay. We obtain okay squared plus 36 equals to nine Y square Plus four G square. This is an ellipse. Now if what he wants to. Okay. Mhm. We obtained that Excess Square -4 G sq Because nine K sq- Statistics. Which is in hi Pamela. And then if G equals to care we obtain that excess square minus nine Y squared equals to four K square minus 36 which is a hyperbolic. So Table one. It follows a decision. Hyperbolic of 16 How do we draw this graph? That is something that we want to figure out. So what I will do is I will draw the sun along the X axis. So let me do that. So what what we know is that access where we want to draw this graph? So first we observed that if I choose any value of X, this is an ellipse. So at every point I really get an ellipse. Yeah. So and the effect is you know I'm on the lips at the X. Y. Plane. So let me draw this is my ex thing, this is lying why by G. So I made out of business by why so much and this is my ex. So if X equals to zero, I'm here on the plane and I get an ellipse. So every point I really get an ellipse. So so what is really happening is that this is something like this and the intersection here is an ellipse for every point it is an ellipse. Yeah. Yeah. Yeah. Mhm. Mhm. Yeah. Yeah. Mhm.

All right. So we want to identify the surface access square plus line. That's why step was 25 plus G squared plus 40 questions. So let's first find it's jess. So, this means that we take extra care. So I obtained y square over 25 because G squared about four is recording one minus K squared of online. But as you can see this is an ellipse. Mhm. On the Y. G. Plan. Let's take the white dress again. Again we take a Y two K. So you get access squared plus nine C. G. Square work fraud Equals to run when his case where over 25 which is again on the lips on the X. Cheap. And similarly, if I take jesus. Mhm. This means that I'll take you to be constant And I would get excess grand around nine plus Why square over 25 is one -K square over source. And this is again an idiots on X. Rite thing. So since all the traces our lips, this isn't a knicks suck so the craft would look like with my chaplain. This is my people in your life will look like it like this. So each. Yes. And if we plot it under Judge Debra, this is how we get right to each partisan election. So the red part is the bottom up and the blue part is a top up. This is how I plotted it to illustrate that this is just a mirror reflection on the top most. But yes.


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