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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Are there any points on the hyperboloid x2y2z2 = 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

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Chapter 14 Solutions

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