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If the level surface of thc functionI16,0,4=}-r 212that passes through the point ( 2,1,4) Intersects the Xz-plane in a curve € , Find the equation of curve â...

Question

If the level surface of thc functionI16,0,4=}-r 212that passes through the point ( 2,1,4) Intersects the Xz-plane in a curve € , Find the equation of curve €

If the level surface of thc function I 16,0,4=}-r 212 that passes through the point ( 2,1,4) Intersects the Xz-plane in a curve € , Find the equation of curve €



Answers

Find the equation of the surface that results when the curve $4 x^{2}+3 y^{2}=12$ in the $x y$ -plane is revolved about the $y$ -axis.

Hello there. So here we have this curve defy on the xy plane and then we need to generate a surface of revolution by revolving it about the X axis. So technically you can see it by the structure of this equation that correspond to high parabola and high prevalence. That is a fine on the xy plane. So technically you can visualize this hyper la here in this plot. These two blue curbs that you can observe that are on the why explain corresponds to four X squared minus three Y square equals to 12. And then you are going to rotate about the X axis. So you're going to rotate in this direction and you see that you generate some how kind of and it took two olympic prov alloys. But in general the surface corresponds to a to to sheet parabolic hyperbolic. Okay, so this surface corresponds to this two sheets hyper blood. So how to obtain the equation of this surface? What it needs to be. It is related to this revolution that you're forming about the X axis. So we need to remain X. Environment because we're revolving about that access. And then we need to change the Y. So we perform the following change of variables why it's going to became Y squared plus Z squared. And then replacing into the equation, we obtain four exit square -3 times wide square plus minus 30 square equals to 12. And then if we spend further we obtain X square over three minus. Why Square over 4-. Here is Z. The square minus Z square over four equals to one. And then you can clearly visualize that this corresponds to the two she ate. Ah Hyper V. Lloyd.

So far. Judge, you find? Ah, intersection Kurt. Region it Jule surface here. So why would you want to come with the reckoning? Is so have sent Echo Jew I mean right in black here. So that ever becomes the angst. Oh, uh less far six thanks. And then minus one. And now we want to find ah Dunton the slumber dental life. So you had your fun isn't Bram he called you for experiment tree the six And now we're introduced a variety And upon Is that echo 26 and thanks Kikuchi. Once we should get the the Bram we could do Ah Far Temp's one about chamber six. When we fall, they go to 10 So does love under bone be here? We could you 10

Well, this problem, we're going to find the equation of the level curve for the given function, passing through the point negative warning to one. So we know that the level curves will have the form natural of X squared plus y, but the square is equal to some kind to find that constant. See, you're going to evaluate the function at the given point. But here we have not your log, uh, negative once where Plus two What? One square And that is simply equal to the natural law of four. The right subset turned that into the equation. We have not your love of ex cleared this one was it? Squared is equal to the natural laws of four. We can simplify this equation buying going into but in the base of e on each of these. So what we end up with is the equation x squared plus y plus, the square is equal four. So this would be the equation for the level curl that goes through the point. Negative one to one

So the given function is f of X Conway Commons. There is equal to natural lower X squared bliss. Weightless That square So Level's offices are obtained by setting F is equal to C. C. Is a constant. So she is you will do. That's a logo. Ex purpose wipers Their square yes Fellows office were given the levels of this passes through minus one comma, comma one substituting bigger. She is able to natural logo physical to natural law before. So a level surface passing through the given point is natural law before is he gonna do natural logo X squared plus y plus X squared is equal in tow. Expert bliss way because they're squared is equal to four. Is the level surface passing through the human point?


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