Question

find and identify the traces of the quadric surface x² y² − z ²= 16 given the plane. x = k find the trace. identify the trace. circle ellipse hyperbola parabola y = k find the trace.

Answer 1

To find the traces of the quadric surface x² + y² - z² = 16 on the planes x = k and y = k, we substitute k for x and y respectively. The resulting equations represent hyperbolas if k² is less than 16. If k² is greater than 16, no real trace exists.

Step-by-step explanation:

The question asks to find and identify the traces of the quadric surface x² + y² - z² = 16 on the planes defined by x = k and y = k. To find the trace on the plane x = k, we substitute k for x in the equation, yielding k² + y² - z² = 16. This is the equation of a hyperbola when k² is less than 16 since the y and z terms have opposite signs. If k² is greater than 16, the equation has no real solution, hence no trace in real space. Similarly, for the plane y = k, we substitute k for y to get x² + k² - z² = 16, with the same reasoning applied. Each trace must then be identified as either a hyperbola or no trace at all, depending on the value of k.