Identify and sketch the surface 9x² + y² – 2z² — 18x - 4y - 12z - 9 = 0.Explain the position and the orientation of the surface.

Answered: Identify and sketch the surface 9x² +…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Identify and sketch the surface 9x² + y² – 2z² — 18x - 4y - 12z - 9 = 0.
Explain the position and the orientation of the surface.

Expert Solution

Step 1

The given surface is:

$9 x^{2} + y^{2} - 2 z^{2} - 18 x - 4 y - 12 z - 9 = 0$ .

The given equation can be written as:

$\begin{array}{rcl} 9 x^{2} + y^{2} - 2 z^{2} - 18 x - 4 y - 12 z - 9 & = & 0 \\ 9 x^{2} - 18 x + y^{2} - 4 y - 2 z^{2} - 12 z - 9 & = & 0 \\ 9 (x^{2} - 2 x) + (y^{2} - 4 y) - 2 (z^{2} + 6 z) - 9 & = & 0 \\ 9 (x^{2} - 2 x + 1 - 1) + (y^{2} - 4 y + 4 - 4) - 2 (z^{2} + 6 z + 9 - 9) - 9 & = & 0 \\ 9 ({(x - 1)}^{2} - 1) + ({(y - 2)}^{2} - 4) - 2 ({(z + 3)}^{2} - 9) - 9 & = & 0 \\ 9 {(x - 1)}^{2} - 9 + {(y - 2)}^{2} - 4 - 2 {(z + 3)}^{2} + 18 - 9 & = & 0 \\ 9 {(x - 1)}^{2} + {(y - 2)}^{2} - 2 {(z + 3)}^{2} & = & 4 \\ \frac{9 {(x - 1)}^{2}}{4} + \frac{{(y - 2)}^{2}}{4} - \frac{2 {(z + 3)}^{2}}{4} & = & 1 \\ \frac{{(x - 1)}^{2}}{{(\frac{2}{3})}^{2}} + \frac{{(y - 2)}^{2}}{2^{2}} - \frac{{(z + 3)}^{2}}{{(\sqrt{2})}^{2}} & = & 1 \end{array}$