   Chapter 12.6, Problem 32E

Chapter
Section
Textbook Problem

# Reduce the equation to one of the standard forms, classify the surface, and sketch it.4x2 − y2 + 2z2 = 0

To determine

To classify: The given surface equation and sketch the surface equation 4x2y+2z2=0

Explanation

Given data:

Surface equation is 4x2y+2z2=0 .

Formula used:

Consider the standard equation of elliptic paraboloid along the y axis.

yb=x2a2+z2c2 (1)

Consider the given surface equation.

4x2y+2z2=0

y=4x2+2z2 (2)

Rearrange the equation.

y1=4x21+2z21 (3)

By comparing equation (3) with (1), the computed expression satisfies the equation of elliptic paraboloid along the y-axis.

Thus, the surface equation 4x2y+2z2=0 is an elliptic paraboloid along the y-axis.

Find a, b, and c by comparing equation (3) with equation (1).

a2=1a=1a=±1

The value of a is ±1 .

b=1

The value of b is 1 .

c2=1c=1c=±1

The value of c is ±1 .

Case i:

Let x=k .

Substitute k for x in equation (2),

y=4k2+2z2

The trace of this expression for x=k represents a family of parabola

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