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

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

\(z=\dfrac{x^{2}}{9}-\dfrac{y^{2}}{4}\)

Solution (a) We begin by dividing the equation by \(9\) to put it in standard form. \[ \dfrac{x^{2}}{9}-\dfrac{y^{2}}{9}+\dfrac{z^{2}}{9}=-1 \]

This is a hyperboloid of two sheets. It has two intercepts \(( 0,3,0)\) and \((0,-3,0).\) If \(y=0\), the equation has no real solution, so there is no trace in the \(xz\)-plane. Traces parallel to the \(xz\)-plane are ellipses and are defined for \(\left\vert y\right\vert \gt3.\) The trace in the \(xy\)-plane is the hyperbola \(\dfrac{y^{2}}{9}-\dfrac{x^{2}}{ 9}=1\) and the trace in the \(yz\)-plane is the hyperbola \(\dfrac{y^{2}}{9}- \dfrac{z^{2}}{9}=1.\) Traces parallel to the \(xy\)- and \(yz\)-plane are also hyperbolas.

The \(y\)-axis is the axis of this hyperboloid of two sheets. See Figure 67 for the graph.

(b) This equation defines a hyperbolic paraboloid. The only intercept is at the origin. The trace in the \(xy\)-plane is the pair of lines \(y=\pm \dfrac{2}{3}x,\) which intersect at the origin. The traces parallel to the \(xy\)-plane are hyperbolas given by the equations \(\dfrac{x^{2}}{9}- \dfrac{y^{2}}{4}=k,\) where \(z=k.\)

The trace in the \(xz\)-plane is the parabola \(z=\dfrac{x^{2}}{9}\), which opens up. Traces parallel to the \(xz\)-plane are also parabolas that open up. The trace in the \(yz\)-plane is the parabola \(z=-\dfrac{y^{2}}{4},\) which opens down. Traces parallel to the \(yz\)-plane are also parabolas that open down.

The origin is the saddle point of the hyperbolic paraboloid. See Figure 68 for the graph.