   Chapter 12.6, Problem 13E

Chapter
Section
Textbook Problem

# Use traces to sketch and identify the surface.x2 = 4y2 + z2

To determine

To identify: The given surface equation and sketch it.

Explanation

Given data:

Surface equation is x2=4y2+z2 .

Formula used:

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

x2a2=y2b2+z2c2 (1)

Consider the given surface equation.

x2=4y2+z2 (2)

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

Case i:

Let x=k .

Substitute k for x in equation (2),

k2=4y2+z2 (3)

Modify equation (3) for k=0 ,

4y2+z2=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

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