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

$ x^2 - y^2 + z^2 - 4x - 2z = 0 $

Standard form: $(x-2)^{2}-y^{2}+(z-1)^{2}=5$

This is an one-sheeted hyperboloid

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So what we want to do is we want to reduce the full language equation which is accessed square minus Y squared. And that's the G squared minus four X. Manage to G0 to the standard form. So we want to complete the square. Yeah. So when you use this technique to complete the square for access by -4 X. Then why school doesn't have any other job? And GT 2 -2 equals. So this implies that access square And there's two times 2 x plus two square Venezuela's square loss G square And as two times 1 times G last for a while. This was since about it four years, a lot of four years. And then I've already won. Here's another one on the right hand side. But this is x minus two square minus y square plus G minus one square is five. So this is the starter. Yes. Now what is this? So this is this is really a hyperbolic weight of 16. This is a hyper below it. Okay. Mhm. And how do we draw it? So this is something this looks like something like this. So I have to supporters at the end of the day and then like this they do not really join. So it just Yeah, something of this. So this is hyperbole. Mhm. Yes, cleansing. Now if we have So what are the other barriers and suffering? There are two other variations. One of them is that there's a conical surface in between. So in that case, what it would look like is something like this and then this is conical surface in between. It's just Mhm. Mhm. The third form is at the hyper bullet of boosters in his case they're completely separated. So something like this. Shut up. Yeah. Yes. Yeah. Right. Alright. So now let us go back to the question so how do we how do we see this better? So we can draw this in a more precise ways. We can say that this is experienced two squared, remember the equation experience to square minus y. Square. Mhm. Plus G -1 Squared equals to fight. So what we have is that effects minus two squared plus g minus one square Is 5-plus rice cash. Not for convenience. Let me draw the Y axis here, he's my Lexus and these are my X. Mhm. This is my he says X says man, she accepts similarly. This comes down to this. Yeah. Yeah. So now So what we do, so remember if why is zero at this point, I'm at this point then it is really uh a circle center at radius too, one common right satu than one year and radius of something like this. It is it is something a circle at this point. Similarly, if I so the center is always remains this year. Now, if I change my wife, if I change my wife to one at the same thing at UAE called one plane, it is again uh circle center at one comma two, but more bigger radius, something like this, and then go on, it goes on increasing. So it's something like something like this is really happening and then something like this, similarly, it goes down as well. Mhm. Yeah, and this is exactly what we have. Yeah, so these are the circles are reforming at each plane up, so this is this is a hyper bullet of once.