   Chapter 14, Problem 31RE

Chapter
Section
Textbook Problem

Find the points on the hyperboloid x2 + 4y2 − z2 = 4 where the tangent plane is parallel to the plane 2x + 2y + z = 5.

To determine

To find: The point on the hyperboloid x2+4y2z2=4 which is the tangent plane parallel to the plane 2x+2y+z=5 .

Explanation

Given:

The equation of the hyperboloid is, x2+4y2z2=4 .

The equation of the plane is 2x+2y+z=5 .

Calculation:

The normal vector to the hyperboloid is, F(x,y,z)=x2+4y2z24 .

F(x,y,z)=Fx,Fy,Fz=x(x2+4y2z24),y(x2+4y2z24),z(x2+4y2z24)=(2x),(8y),(2z)

Thus, normal vector of the hyperboloid is, F(x,y,z)=2x,8y,2z .

The plane equation is, 2x+2y+z=5 .

The tangent plane is parallel to the plane equation 2x+2y+z=5 only if the normal vectors of the planes are parallel. Thus, find the point on the hyperboloid such that F(x,y,z)=k2,2,1

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