Identify each quadric surface. List its intercepts and its traces in the coordinate planes.
The intercepts of the ellipsoid are (3,0,0), (−3,0,0), (0,2,0), (0,−2,0), (0,0,6), and (0,0,−6).
The traces are all ellipses. In the xy-plane, the trace is x29+y24=1; in the xz-plane, the trace is x29+z236=1; and in the yz-plane, the trace is y24+z236=1. Figure 62 shows the graph of the ellipsoid.
(b) This equation defines an elliptic paraboloid. Its only intercept (vertex) is (0,0,0).
The trace in the yz-plane is the vertex (0,0,0) and traces parallel to the yz-plane are ellipses, provided x>0.
To find the trace in the xy-plane, let z=0. The trace is the parabola x=y24. To find the trace in the xz-plane, let y=0. The trace is the parabola x=z2.