Question

(a) Find and identify the traces of the quadric surface

x² + y² ? z² = 16

given the plane.

x = k

Find the trace.

y = k

Find the trace.

z = k

Find the trace.

Describe the surface from one of the graphs in the table.

ellipsoid

elliptic paraboloid

hyperbolic paraboloid

cone

hyperboloid of one sheet

hyperboloid of two sheets

Answer 1

Given quadratic is $x^2+y^2-z^2=16$

Given x=k

So $k^2+y^2-z^2=16$

$y^2-z^2=16-k^2$

Here k can be any number.

So while tracing let us look at the possibilities.

if k=+ - 4, the graph would represent two lines only.

At all other values this represents a hyperbola.

It is same with y=k

But for z=k

$x^2+y^2=16+k^2$

So this is a circle with radius of $\sqrt{16+k^2}$

Finally the surface altogether is a set of circle which in other directions is hyperbola.

So it is a hyperboloid on one sheet.