   Chapter 14, Problem 28RE

Chapter
Section
Textbook Problem

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.28. xy + yz + zx = 3, (1, 1, 1)

(a)

To determine

To find: The equation of the tangent planeto the surface xy+yz+zx=3 at the point (1,1,1) .

Explanation

Given:

The surface is, xy+yz+zx=3 .

Result used:

“The tangent plane to the level surface at the point P(x0,y0,z0) is defined as Fx(x0,y0,z0)(xx0)+Fy(x0,y0,z0)(yy0)+Fz(x0,y0,z0)(zz0)=0

Calculation:

Let the surface function be, F(x,y,z)=xy+yz+zx3 (1)

The equation of the tangent plane to the given surface at the point (1,1,1) is defined by,

Fx(1,1,1)(x1)+Fy(1,1,1)(y1)+Fz(1,1,1)(z1)=0 (2)

Take partial derivative with respect to x at the point (1,1,1) in the equation (1),

Fx(x,y,z)=x(xy+yz+zx3)=y+0+z0=y+zFx(1,1,1)=2

Thus, the value of Fx(1,1,1)=2

(b)

To determine

To find: The equation of the normal line to the surface xy+yz+zx=3 at the point (1,1,1) .

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