Describe the trace of the surface given by the function below in the xy-plane. x² + y² + z² = 1 A) parabola B) ellipse (not circle) C) circle D) hyperbola E) line

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Quizplus · Accepted Answer

Rewriting the equation in standard form can give us insight into the shape of the surface:
x^2 + y^2 + z^2 = (1/3)(x+y+z)^2
This is the equation of an ellipsoid centered at the origin with semi-axes of length 1, 1, and sqrt(3). When this surface is projected onto the xy-plane (by setting z=0), we get an ellipse with major axis of length 2 and minor axis of length 2/sqrt(3). Therefore, the trace of the surface in the xy-plane is an ellipse (not a circle).

Describe the Trace of the Surface Given by the Function