Question

(a) Find and identify the traces of the quadric surface x2 + y2 − z2 = 81 given the plane. x = k Find the trace.

Answer 1

The trace of the quadric surface x^2 + y^2 - z^2 = 81 given the plane x = k is a circle with radius 9 and center at (k, 0, 0).

Step-by-step explanation:

The given quadric surface is x2 + y2 - z2 = 81.

To find the traces of the surface given the plane x = k, we substitute k for x in the equation of the surface.

The trace is a curve formed by the intersection of the surface and the plane. In this case, the trace is a circle with radius 9 and center at (k, 0, 0).

Answer 2

Hyperbola

Step-by-step explanation:

Consider
`x=k.`

Substitute
`x=k` in given equation
`x^2+y^2-z^2=81`

`k^2+y^2-z^2=81`

`\Rightarrow y^2-z^2=9^2-k^2`

`\Rightarrow y^2-z^2=(9-k)(9+k)`

Here, different orientation for
`-9<k<9` then
`-9<k` or
`k<9.`

Hence, the surface equation represents a trace of the hyperbola.