>> To set up the next problem, imagine that I have a basketball. Imagine that I'm holding a basketball and I take this basketball and I lower it into a tub of water. And let's say that I just kind of push it a little bit into the tub of water to where it's just maybe oh one-quarter submerged or something like that. Now imagine where as the water meets the sphere, as the water meets the basketball, what kind of figure is formed? The interception of the water with the basketball right at the surface of the water. You see it's just a circle. It's just a circular path for the water, and that's what we're talking about in the next problem and that path is called a trace. Now we're thinking here about finding a trace in a particular plane associated with the graph of this sphere. Now this sphere has a center at 3, 2 negative 4 identified here, and a radius of 5, and we want the X, Y trace. Now that's the trace sort of the intersection then of the sphere with the X Y plane. That's what the X Y trace is and it's going to form a circle. So what's going to emerge here will be the equation for a circle for that X Y trace. And all we do to identify a trace is to understand that for all of the values, the coordinates or points that lie on the X Y plane. What are the characteristics of the coordinates of those points? Well every one of those points that lies on the X Y plane would have a Z component, a Z coordinate of 0. So the distance from the X Y plane is 0 if we're on the X Y plane. That's the idea so the Z value is 0 so all we have to do is take the equation up here and replace the Z with 0 you see and evaluate. Now that just causes us to change the value of the constant over here and the Z term goes out. But we have 0 plus 4, 4 squared is 16, if I subtract 16 on both sides, I get 9 for the constant on the right; and these components stay the same. So the trace is a circle and here is its equation, and the circle has a center at 3, 2 and it has a radius of 3. We could think about the X Z trace, the trace or the sort of the imprint that this graph would make, this sphere would make in the X Y plane, you see, the intersection of those too. Now think about that--in the X Z plane the Y value would be 0 so all we would have to do is replace Y with 0 and evaluate in a way that is similar to this one.