Quadric Surface and the traces

Define a quadric surface:

surface = 16x^2-y^2-4z^2

Here is the trace in the planes z=k. This is the equation of a family of hyperbolas if k≠0, and two straight lines if k=0. Notice the "upper" and "lower" parts are not included; the hyperbolas are only to the "left" and "right". The traces parallel to the other planes follow, and we see that we have ellipses parallel to the yz-plane (x=k) and hyperbolas parallel to the xz-plane (y=k).

trace=surface==0/.z->k
listtrace=Table[trace/.k->n,{n,-4,4}]
ContourPlot[Evaluate[listtrace],{x,-2,2},{y,-5,5}]

trace=surface==0/.x->k
listtrace=Table[trace/.k->n,{n,-5,5}]
ContourPlot[Evaluate[listtrace],{y,-10,10},{z,-10,10}]

trace=surface==0/.y->k
listtrace=Table[trace/.k->n,{n,-5,5}]
ContourPlot[Evaluate[listtrace],{x,-2,2},{z,-5,5}]

Here is the quadric surface in three dimensions:

surfaceplot = ContourPlot3D[surface,{x,-5,5},{y,-5,5},{z,-5,5},Contours->0, Axes->True,AxesLabel->{"x-axis","y-axis","z-axis"}]