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

\(4x^{2}+9y^{2}+z^{2}=36\)

\(x=\dfrac{y^{2}}{4}+z^{2}\)

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. \[ \begin{eqnarray*} \dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}+\dfrac{z^{2}}{36} &=&1 \\ \dfrac{x^{2}}{3^{2}}+\dfrac{y^{2}}{2^{2}}+\dfrac{z^{2}}{6^{2}} &=&1 \end{eqnarray*} \]

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 \(\dfrac{x^{2}}{9} +\dfrac{y^{2}}{4}=1;\) in the \(xz\)-plane, the trace is \(\dfrac{x^{2}}{9}+ \dfrac{z^{2}}{36}=1;\) and in the \(yz\)-plane, the trace is \(\dfrac{y^{2}}{4}+ \dfrac{z^{2}}{36}=1.\) 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 \(x\gt0\).

To find the trace in the \(xy\)-plane, let \(z=0\). The trace is the parabola \(x= \dfrac{y^{2}}{4}.\) To find the trace in the \(xz\)-plane, let \(y=0\). The trace is the parabola \( x=z^{2}\).