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: 17


Hyperboloid of one sheet; the trace is a hyperbola in the $ xz $-plane.

Work Step by Step

The equation $$ \left(\frac{x}{3}\right)^{2}+ \left(\frac{y}{5}\right)^{2} -5z^2=1 $$ can rewritten as follows $$ \left(\frac{x}{3}\right)^{2}+ \left(\frac{y}{5}\right)^{2} -\left(\frac{z}{\sqrt 5}\right)^{2}=1 $$ which is a hyperboloid of one sheet. (See equations on page 691.) To find the trace with the plane $ y=1$, we have $$ \left(\frac{x}{3}\right)^{2}-\left(\frac{z}{\sqrt 5}\right)^{2}=\frac{24}{25} $$ which is a hyperbola in the $ xz $-plane.
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