Use traces to sketch and identify the surface.
$ x^2 = 4y^2 + z^2 $
All right. They want us to use traces to figure out what this is a picture of. So first um let zb zero. So we're looking in the xy plane? So X squared equals X squared equals four Y squared. So x equals plus and -2 y. Or if it's easier for you to think about why equals plus minus two X. A half X. Sorry. Okay. So those are lines in the xy plane. Sex Y plane go through 00, slope is one half And -1 half. Okay. And then um in let Y equals zero. And then you get X squared equals Z squared. So X. Or Z equals plus minus X. So in the Z explain also lines but steeper. Here's excuse the these have slope one and -1. Oh okay. So then what about in the other plane you get uh let X equal zero. And so you get four Y squared plus Z squared equals zero. Okay so that's not happening Unless X&Y are both zero. Okay. Um So that's the zero. But how about if for example X. Is one then one squared equals four Y squared plus B squared. So that's any lips in the Y. Z plane um One squared equals y squared over 1/4. Was he squared? So it's like this. Alright let's put it all together. So for sure it goes to this point right here um And the in the X. What in the Wednesday was zero? So in the xy plane, its lines in the xz plane? Its lines. And in the Y Z. Plane it's ellipses. Okay, so in the which which way did the ellipses go? Um I got to look again fat. Tall on the z. Smaller on the line. Yeah. Okay. So here's here's a line in the XZ plane. Here's a line in the Xz plane. Here's a line in the UAE explain. Here's a line in the UAE explain. I kind of got a shifted a little bit. Okay, so it's a cone, it opens on the X. Axis on the positive X. Axis and it's not a circular comb but uh elliptical cone, that's um 1-1 half here or twice as big as twice as tall as it is wide. Okay, so here's how I got that. It was this that told me what to do. I got to have ellipses in the Y. Z. Plane. Okay. Button or parallel to the Y. Z. Plane? In the Y. Z. Plane. Just have a point. And behind the Y. Z. Plane, we don't have anything but coming out of the Y. Z. Plane, we have ellipses. All right,