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

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

elliptic paraboloid

Vectors

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All right. We want to write the standard form of access square, man. That's why Has to this curricula zero. So then we just move right to the other side. So I get why he was 24 X squared plus two G square. This is this is the standard form and this is an elliptic parabola. It and the reason is the following. So if I see that, why cannot be negative? So this is really on the I want you to start up boy and that for any value of Y or any Why value we call this value. Okay, what we get is that four X squared plus two G square equals two K. Is an ellipse on the X G plan. Mm hmm. So this means that at every value of K we get ellipse and the value of cable increases the the dimension population increases. So, so what we get is a parabola. Well, joining these ellipses and let me draw this for the sake of convenience. Again, let me draw the Y axis on the top. And this is my ex cheap. So like I said, why value is always positive? So this is a positive value and or what we call the zero, my X. And year zero. Then from what I got one I get I instead of a circle, I get an ellipse like this. Similarly for why equal to two, I get a bigger lips And what I call the three I got a bigger bigger election. So so if I join all of this, so you get something like, so this is an elliptic parabolas. So each if you if you draw an intersection, if you draw a plane here, this plane intersects this elliptic parabola. It's what we get in the intersection is an ellipse. This is exactly what is up.