Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 12 - Review - Concept Check - Page 841: 19

Answer

The six types of quadratic surfaces are elliptic paraboloid, hyperbolic paraboloid, cone, ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets. Elliptic paraboloid: $\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y}{b}=\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x}{a}=\frac{y^2}{b^2}+\frac{z^2}{c^2}$ Hyperbolic paraboloid: $\frac{z}{c}=\pm\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y}{b}=\pm\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x}{a}=\pm\frac{y^2}{b^2}+\frac{z^2}{c^2}$ Cone: $\frac{z^2}{c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y^2}{b^2}=\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x^2}{a^2}=\frac{y^2}{b^2}+\frac{z^2}{c^2}$ Ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ Hyperboloid of one sheet: $-\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ Hyperboloid of two sheets: $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ or, $-\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

Work Step by Step

The six types of quadratic surfaces are elliptic paraboloid, hyperbolic paraboloid, cone, ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets. Elliptic paraboloid: $\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y}{b}=\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x}{a}=\frac{y^2}{b^2}+\frac{z^2}{c^2}$ Hyperbolic paraboloid: $\frac{z}{c}=\pm\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y}{b}=\pm\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x}{a}=\pm\frac{y^2}{b^2}+\frac{z^2}{c^2}$ Cone: $\frac{z^2}{c^2}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ or $\frac{y^2}{b^2}=\frac{x^2}{a^2}+\frac{z^2}{c^2}$, or $\frac{x^2}{a^2}=\frac{y^2}{b^2}+\frac{z^2}{c^2}$ Ellipsoid: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ Hyperboloid of one sheet: $-\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ Hyperboloid of two sheets: $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ or, $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ or, $-\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
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