Final answer:
The trace of the quadric surface x² - y² - z² = 3 in the XY-plane and XZ-plane is a hyperbola, but there is no real trace in the YZ-plane as the result would not yield real number solutions.
Step-by-step explanation:
The question asks to find and identify the trace of the quadric surface x² - y² - z² = 3. A trace is a curve obtained by intersecting a three-dimensional surface with a plane.
In this case, we can find traces of the given quadric surface by setting one variable at a time to zero and solving for the other two variables, resulting in equations of two-dimensional conic sections.
For example, to find the trace in the XY-plane, we set z=0, and the equation becomes x² - y² = 3. This represents a hyperbola in the XY-plane because one square term is positive and the other is negative.
Similarly, setting x=0, we get -y² - z² = 3, which has no real solutions since the left side cannot be positive. And by setting y=0, we get x² - z² = 3, another hyperbola, but this time in the XZ-plane.