   Chapter 12, Problem 31RE

Chapter
Section
Textbook Problem

# Identify and sketch the graph of each surface.31. x2 = y2 + 4z2

To determine

To identify: The given surface equation and sketch it.

Explanation

Given data:

Surface equation is x2=y2+4z2 .

Formula used:

Consider the standard equation of elliptic cone along the x axis.

x2a2=y2b2+z2c2 (1)

Consider the given surface equation.

x2=y2+4z2 (2)

By comparing equation (2) with (1), the computed expression satisfies the equation of an elliptic cone.

Therefore, the surface equation x2=y2+4z2 is a cone.

Case i:

Let x=k .

Substitute k for x in equation (2),

k2=y2+4z2 (3)

Modify equation (3) for k=0 ,

y2+4z2=0

The integer solution of this equation is,

(y,z)=(0,0)

Hence, the surface equation has a origin point for k=0 .

Similarly, trace of the surface equation is empty for k<0 and the surface equation have elliptical trace for k>0 .

Case ii:

Let y=k .

Substitute k for y in equation (2),

x2=k2+4z2x2=4z2+k2

The trace of this expression represents a hyperbola for k0 and has a two intersecting lines for k=0

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