Find equations for the tangent plane and the normal line to the surface with equation $x y + y z + x z = 38$ at the point $P ( 2,4,5 )$
A) $6 x + 7 y + 9 z = 76$ , $\frac { x - 2 } { 6 } = \frac { y - 4 } { 7 } = \frac { z - 5 } { 9 }$
B) $9 x + 7 y + 6 z = 76$ , $\frac { x - 2 } { 9 } = \frac { y - 4 } { 7 } = \frac { z - 5 } { 6 }$
C) $2 x + 4 y + 5 z = 76$ , $\frac { x - 6 } { 2 } = \frac { y - 7 } { 4 } = \frac { z - 9 } { 5 }$
D) $2 x + 4 y + 5 z = 76$ , $\frac { x - 9 } { 2 } = \frac { y - 7 } { 4 } = \frac { z - 6 } { 5 }$

Question

Quizplus · Accepted Answer

First, we find the gradient vector of the surface at point $P$:
$$\nabla f(x,y,z) = \langle y+z, x+z, x+y\rangle$$
$$\nabla f(2,4,5) = \langle 9,7,6\rangle$$

This gradient vector is normal to the surface at point $P$. So, the equation of the tangent plane is:
$$9(x-2)+7(y-4)+6(z-5)=0$$
Simplifying, we get:
$$9x+7y+6z=76$$
So, option B is the correct choice for the equation of the tangent plane.

Next, we use the fact that the normal line to the surface at point $P$ is parallel to the gradient vector at that point. So, the equation of the normal line is of the form:
$$\frac{x-2}{a}=\frac{y-4}{b}=\frac{z-5}{c}$$
where $\langle a,b,c\rangle$ is the gradient vector at point $P$. Substituting the values, we get:
$$\frac{x-2}{9}=\frac{y-4}{7}=\frac{z-5}{6}$$
So, option B is the correct choice for the equation of the normal line.

Find Equations for the Tangent Plane and the Normal Line xy+yz+xz=38x y + y z + x z = 38xy+yz+xz=38

Find Equations for the Tangent Plane and the Normal Line $x y + y z + x z = 38$