Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 13 - Vector Geometry - 13.6 A Survey of Quadratic Surfaces - Exercises - Page 692: 18


Hyperboloid of two sheets; the trace is an ellipse in the $ xy $-plane.

Work Step by Step

The equation $$ 4x^2+ \left(\frac{y}{3}\right)^{2} -2z^2=-1 $$ can rewritten as follows $$ \left(\frac{x}{1/2}\right)^{2}+ \left(\frac{y}{3}\right)^{2} =\left(\frac{z}{1/\sqrt 2}\right)^{2}-1 $$ which is a hyperboloid of two sheets. (See equations on page 691.) To find the trace with the plane $ z=1$, we have $$ \left(\frac{x}{1/2}\right)^{2}+ \left(\frac{y}{3}\right)^{2} =1 $$ which is an ellipse in the $ xy $-plane.
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