Answered: Write in standard form and identify the…

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

Write in standard form and identify the surface. Use at least two traces and sketch. Label the interesting points and try to be neat:

6x²+y+24z²=0

Expert Solution

Step 1

Given,

The surface, $6 x^{2} + y + 24 z^{2} = 0$

Step 2

Consider the given quadratic surface, $6 x^{2} + y + 24 z^{2} = 0$

It can be rewritten as,

$6 x^{2} + 24 z^{2} = - y \Rightarrow x^{2} + \frac{24}{6} z^{2} = - \frac{1}{6} y \Rightarrow x^{2} + 4 z^{2} = - \frac{1}{6} y \Rightarrow \frac{x^{2}}{4} + \frac{z^{2}}{1} = - \frac{1}{6} y \Rightarrow \frac{x^{2}}{2^{2}} + \frac{z^{2}}{1^{2}} = - \frac{1}{6} y$

The standard form of Elliptic paraboloid is $\frac{x^{2}}{a^{2}} + \frac{z^{2}}{b^{2}} = c y$ .

Thus, the given quadratic surface is elliptical paraboloid.

Step 3

Consider, the given surface $6 x^{2} + y + 24 z^{2} = 0$

The parabola opens along the variable with degree 1 and its coefficient gives the direction.

Here, parabola is opens along y axis with -1/6 coefficient, so it is along negative y axis.

Set x= 0, gives $24 z^{2} = - y$

This is the parabola along negative y axis.

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