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Identify each quadric surface. List its intercepts and its traces in the coordinate planes.

  • 4x2+9y2+z2=36
  • x=y24+z2
  • Solution (a) We recognize this as an ellipsoid because it has three squared terms and all the coefficients are positive. To find its intercepts and traces, divide the equation by 36 to put it in standard form. x29+y24+z236=1x232+y222+z262=1

    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.