INSTRUCTOR: A hyperboloid of one sheet is any surface that can be described with an equation of the form x squared over a squared plus y squared over b squared minus z squared over c squared equals 1. Describe the traces of the hyperboloid of one sheet given by the equation x squared over 3 squared plus y squared over 2 squared minus z squared over 5 squared equals 1.

Now how we'll describe this by projecting into each of these planes is we will simply let some values be 0. So for the xy plane, we'll let the z component be 0. And if we let the z component be 0, that would be x squared over 3 squared plus y squared over 2 squared equals 1, which you may notice is an ellipse.

That is an ellipse. We're just looking for shapes here, so I'm describing this. So we have an ellipse in the xy plane.

For the xz plane, we'll let y be equal to 0. This equation would be x squared over 3 squared minus z squared over 5 squared equals 1, which is going to be a hyperbola. So we have a hyperbolic shape there.

In the yz plane, we'll let the x component be 0. So this resulting equation-- be y squared over 2 squared minus z squared over 5 squared equals 1. And again, we have a hyperbolic shape.

So now I'm going to attempt to draw this in three dimensions. But again, I'm going to use these traces. So in my xy plane, I will have an ellipse-- elliptical shape. In the xz plane, I'm going to have a shape of a hyperbola. Perspective drawing's not so great. But then also from the yz, we're going to have that similar shape.

So let's see if I can draw this better over here, given what a graphing utility would show. Let's see-- a few circles here. So it's almost like a tube that's being squeezed in the middle. So the hyperbolic shape kind of comes out, in that that tube has a slimmer area in the center where you'd be squeezing it, but it follows a hyperbolic shape on the sides.