Answered: (a) Find and identify the traces of the…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.1: The Rectangular Coordinate System

Problem 44E: By definition, a hyperbola is the locus of points whose positive difference of distances from two...

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(a) Find and identify the traces of the quadric surface

x² + y² − z² = 81

given the plane.

x = k

Find the trace.

Identify the trace.

circleellipse hyperbolaparabola

y = k

Find the trace.

Identify the trace.

circleellipse hyperbolaparabola

z = k

Find the trace.

Identify the trace.

circleellipse hyperbolaparabola

Describe the surface from one of the graphs in the table.

ellipsoidelliptic paraboloid hyperbolic paraboloidconehyperboloid of one sheethyperboloid of two sheets

(b) If we change the equation in part (a) to

x² − y² + z² = 81,

how is the graph affected?

The graph is rotated so that its axis is the x-axis.The graph is rotated so that its axis is the y-axis. The graph is rotated so that its axis is the z-axisThe graph is shifted one unit in the negative y-direction.The graph is shifted one unit in the positive y-direction.

x² + y² + 2y − z² = 80?

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Hello. Since your question has multiple parts, we will solve first question for you. If you want remaining sub-parts to be solved, then please resubmit the whole question and specify those sub-parts you want us to solve.

Consider the given:

$\begin{array}{l} x^{2} + y^{2} - z^{2} = 81 \\ Plane x = k \\ k^{2} + y^{2} - z^{2} = 81 \\ y^{2} - z^{2} = 81 - k^{2} \\ This is required trace. \\ The trace is hyperbola. \end{array}$

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