## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 13 - Vector Geometry - 13.6 A Survey of Quadratic Surfaces - Exercises - Page 692: 22

#### Answer

(a) ellipsoid. (b) hyperboloid of one sheet. (b) hyperboloid of two sheets.

#### Work Step by Step

(a) When all signs are "+", we get $$\left(\frac{x}{1/\sqrt 8}\right)^{2}+\left(\frac{y}{1/\sqrt 3}\right)^{2}+z^{2}=1$$ which is an ellipsoid. (b) When two signs are "+", we get $$\left(\frac{x}{1/\sqrt 8}\right)^{2}+\left(\frac{y}{1/\sqrt 3}\right)^{2}-z^{2}=1$$ which is a hyperboloid of one sheet. (b) When one sign is "+", we get $$\left(\frac{x}{1/\sqrt 8}\right)^{2}-\left(\frac{y}{1/\sqrt 3}\right)^{2}-z^{2}=1$$ which is a hyperboloid of two sheets. (See equations on page 691-692.)

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