   Chapter 11.6, Problem 41E

Chapter
Section
Textbook Problem

# Analyzing a TraceIn Exercises 43 and 44, analyze the trace when the surface z = 1 2 x 2 + 1 4 y 2 is intersected by the indicated planes.Find the lengths of the major and minor axes and the coordinates of the foci of the ellipse generated when the surface is intersected by the planes given by(a) z = 2 and (b) z = 8.

(a)

To determine

To calculate: For the gives surface z=x22+y24 find the lengths of major and minor axes and coordinates of foci of the ellipse.

Explanation

Given:

The surface

z=x22+y24

is intersected by the plane

z=2

Formula used:

The equation of ellipse is given by

x2a2+y2b2=1

And, c=b2a2

where, c is the distance from the focus to the vertex of ellipse.

Calculation:

the plane

z=2

Intersects the surface

z=x22+y24

Hence, substitute z=2 in the equation of given surface

(b)

To determine

To calculate: For the gives surface z=x22+y24 find the lengths of major and minor axes and coordinates of foci of the ellipse.

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