Exploring the Impact of Z on Quadratic Surfaces: Hyperbolas in the XY Plane

[nameVoid]

4x^2-y^2+2z^2+4=0
x^2-y^2/4+z^2/2+1=0
-x^2+y^2/4-z^2/2=1

In the xy trace -x^2+y^2/4=1+k^2/2 taking k=0 will yield the hyperbola but what affect will z have on the resulting surface as it tends to +- infinity
It appears to me that as z to +- infinity the hyperbola in the xy plane becomes wider and this is not the case in the graph

 

[tiny-tim]

hi nameVoid!

(try using the X² button just above the Reply box)

-x²+y²/4-z²/2=1

In the xy trace -x²+y²/4=1+k²/2 taking k=0 will yield the hyperbola but what affect will z have on the resulting surface as it tends to +- infinity

k = 0 gives you the "horizontal" slice at z = 0

k = k gives you the general "horizontal" slice at z = k

so (for constant k) what is the shape of -x²+y²/4=1+k²/2 ?

 

[nameVoid]

I'm plotting a few points and the change in y as x changes from 0 to 1 is less as z increases causing the hyperbola to be wider although in the resulting shape it appears to be narrowing

The slice at z=0 should be the widest slice however at this point it has the greatist change in y as xbfrom 0 to 1

Mathematica shows graphs as z becomes large to be within the former this is not the obvious case given the change pattern in y from x 0 to 1 but as z becomes large it looks to be less

As becomes large the hyperbola must widen, although it is not as wide as the 0 cut it still must widen at slightly fast rate because if it's position with respect to x

 

[tiny-tim]

what about the asymptotes?

what does the 3D graph of the asymptotes look like?