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Numerade Educator

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Problem 38 Medium Difficulty

Reduce the equation to one of the standard forms, classify the surface, and sketch it.

$ 4x^2 + y^2 + z^2 - 24x - 8y + 4z + 55 = 0 $

Answer

Standard form: $4(x-3)^{2}+(y-4)^{2}+(z+2)^{2}-1=0$
This is an ellipsoid

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Video Transcript

first thing I noticed when I look at this is that we have an X squared A Y squared in a Z squared. And they are all positive. So it's uh either a sphere or any lips. But since one of them has a number and two of them don't a coefficient, I mean then that tells me it's any lips. If it was a sphere, they would have to all have the same coefficient. All right. So first thing you do is you get all the X terms together, all the white terms together and the z terms together, Okay, because what you're gonna want to do is complete the square. Okay, go ahead and move the 55 over to the other side. Okay? And the first one you can't complete the square with the four there. So you're gonna have to factor it out. So that gives you x squared minus six. Yeah minus six X plus something that you're going to add there. Well, you're gonna have to add the same thing to the other side and remember what yet? It is not just that thing, but it's four times that. All right. And then the same thing I'm y squared. So you're just going to add and then same thing on c squared. Alright, half of negative six is negative three negative three squared is nine. So you added four times 9 to both sides. four times a guy, Half of -8 is -4 -4 score to 16. So, I had 16 of Assad, half of four is 22 squared is four affordable sides. So now this factors into four X. Whatever this nine was before he squared it or whatever is half of minus six, so minus three squared Plus Y -4 Squared. Okay. Plus z plus two squared equals 36 Plus 16. That's 46, plus four. That's 56 Plus -55. So one there. Sure. All right, so then to get it in standard form, you can't have a constant out in the front but multiplying by force, the same thing as dividing by 1/4. Okay, so it's in the lips and then here's its picture. Okay, go to exit. Sorry, center is 34 -2 12312, 3, 4 -2. So there's this center in the X direction, go one third, uh front and back one third. And then in the Y Direction one. Then I think the direction one. All right, so it's kind of a huge center there. All right. Oh, I was saying a lip side. So what I meant was ellipse. I was saying the lips I meant the flip side because it's a three dimensional thing, sort of a football looking thing. There you go