   Chapter 14.6, Problem 55E

Chapter
Section
Textbook Problem

Are there any points on the hyperboloid x2 − y2 − z2 = 1 where the tangent plane is parallel to the plane z = x + y?

To determine

To find: The point on the hyperboloid x2y2z2=1 which is the tangent plane parallel to the plane z=x+y .

Explanation

Given:

The equation of the hyperboloid is, x2y2z2=1 .

The equation of the plane is, z=x+y .

Calculation:

The normal vector to the hyperboloid is, F(x,y,z)=x2y2z21 .

F(x,y,z)=Fx,Fy,Fz=x(x2y2z21),y(x2y2z21),z(x2y2z21)=(2x),(2y),(2z)

Thus, normal vector of the hyperboloid is F(x,y,z)=2x,2y,2z .

The plane equation can be rewritten as, x+yz=0 .

The tangent plane is parallel to the plane equation x+yz=0 only if the normal vectors of the planes are parallel. Thus, find the point on the hyperboloid such that F(x,y,z)=k1,1,1

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Evaluate the definite integral. 03dx5x+1

Single Variable Calculus: Early Transcendentals, Volume I

Find an equation of the line that passes through the point (2, 2) and is parallel to the line 2x 4y 8 = 0.

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Exercises 17 to 22, factor completely. 6y2-54

Elementary Geometry For College Students, 7e

True or False: converges.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

The length of the curve given by x = 3t2 + 2, y = 2t3, is:

Study Guide for Stewart's Multivariable Calculus, 8th 