Final Answer:
a. The trace parallel to the xy-plane (z=0) is a hyperbola, the trace parallel to the xz-plane (y=0) is a parabola opening downward, and the trace parallel to the yz-plane (x=0) is also a parabola opening downward.
b. In R³, the surface is a hyperbolic paraboloid with its axis along the y-axis. It opens downward along the x-axis and upward along the z-axis.
c. The identified surface is a hyperbolic paraboloid.
Step-by-step explanation:
a. The trace parallel to the xy-plane (z=0) is obtained by setting z=0 in the equation. This gives (0 = 1 + y² - x²), which rearranges to (x² = 1 + y²). This is a family of upward and downward opening parabolas along the x-axis. Similarly, setting y=0 and x=0 in the equation gives traces along the xz-plane and yz-plane, respectively. These traces reveal the nature of the surface in each plane.
b. In R³, the full surface is obtained by considering the entire equation (z = 1 + y² - x²). This is a hyperbolic paraboloid, as it contains both parabolic and hyperbolic cross-sections. The axis of the paraboloid is along the y-axis, as the coefficients of y² and -x² are positive and negative, respectively.
c. The identified surface is a hyperbolic paraboloid, which is a quadric surface defined by a second-degree equation. The general form of a hyperbolic paraboloid is (z = ay² - bx²), and in this specific case, (a = 1) and (b = 1), indicating that the surface is symmetric with respect to both the xz and yz planes.