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

$ x^2 + y^2 - 2x - 6y - z + 10 = 0 $

Standard form: $(x-1)^{2}+(y-3)^{2}-z=0$

This is an infinite paraboloid

Vectors

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Okay, so this question asked you to reduce the equipment and this aquarian is X square plus Y square minus two. X -26 Y. In the minus leath plus 10 Equal to zero. So the way you actually standardize it is to combine all the X together and all the while together. So first we move X together. Why together? And the Mhm. So from observation is the X square minus two X. Actually we can convert them to acts square. Uh Sorry to a kind of interesting thing like X square minus two X. And plus one and then minus one from this one, we can actually convert them to why square minus six, Y plus nine and the minus night minus Z Plus 10 and equals to zero. So now we know this part can be converted to X -1 -1 square minus one. And this part can be converted to why miners three square and minus nine minus Z Plus 10 equals to zero. So then it becomes X -1 square plus why minus three Square -Z equals to zero. So then we simplified and this is the answer. Uh the standard form of the surface. So by looking at this, it's actually you can see it's actually very similar to the forms of the equals to a -1 sq why square. And this is actually the surface of the what we call the circular peril. Yeah, for Lloyd surface, that's the surface name of the services. And the only difference here is actually the minimum. So here it's probably a little bit hard to draw the grass that's try so this is the X. Direction. This is why direction. This is the direction. So from this form we know actually when X equal to 1 1 equals 2 3 Good three Z. is that zero place? So it kind of looks like and we can say it should actually move towards so it actually looks like that. Yeah. Mhm Yeah. Mhm. This is a surface. Mhm. And this bottom is actually 130. So it's a little bit hard to generalize. But what we can do is that we put in the two D graph for example if we're owning look at the um why is he plant? We know that this projection will be very like Probably this is a three. Probably like this. This is what we see shape. If we will prove them onto a acsi direction And very similar this is one and it goes like this. If we put them on the xy plan xy plan so what we see is just a circle. So it's actually equals two one and sweet. Mhm. Actually equals circle goes to infinite and this is the answer