Question

Consider the surface x = y²+ z²/9

a) Give the trace in the xy-plane. What is this object?
b) Give the trace in the xz-plane. What is this object?
c) Give the trace in the yz-plane. What is this object?

Answer 1

The trace in the xy-plane is a parabola, the trace in the xz-plane is a sideways parabola, and the trace in the yz-plane is a circle.

Step-by-step explanation:

a) Trace in the xy-plane: To find the trace in the xy-plane, we set z=0. Substituting this into the equation x=y²+z²/9, we get x=y². This represents a parabola in the xy-plane.

b) Trace in the xz-plane: To find the trace in the xz-plane, we set y=0. Substituting this into the equation x=y²+z²/9, we get x=z²/9. This represents a sideways parabola in the xz-plane.

c) Trace in the yz-plane: To find the trace in the yz-plane, we set x=0. Substituting this into the equation x=y²+z²/9, we get y²+z²/9=0. This represents a circle with radius 3 in the yz-plane.