Find equations for the tangent plane and the normal line to the surface with equation $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$ at the point $P ( 2,1,1 )$.
A)
$x + 9 y + 9 z = 22 , \frac { x - 2 } { 2 } = y - 9 = z - 9$
B)
$x + 9 y + 9 z = 22 , \frac { x - 2 } { 2 } = \frac { y - 1 } { 9 } = \frac { z - 1 } { 9 }$
C)
$2 x + 9 y + 9 z = 22 , \quad \frac { x - 2 } { 2 } = \frac { y - 1 } { 9 } = \frac { z - 1 } { 9 }$
D)
$$2 x + 9 y + 9 z = 22 , \frac { x - 2 } { 2 } = y - 9 = z - 9$$

Question

Quizplus · Accepted Answer

The gradient of the surface at point $P(2,1,1)$ is $(2x, 18y, 18z)$. At $P$, it becomes $(4, 18, 18)$. The tangent plane equation is $4(x-2) + 18(y-1) + 18(z-1) = 0$, simplifying to $2x + 9y + 9z = 22$. The normal line uses the direction vector $(4, 18, 18)$, giving $\frac{x-2}{2} = \frac{y-1}{9} = \frac{z-1}{9}$.

Find Equations for the Tangent Plane and the Normal Line x2+9y2+9z2=22x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22x2+9y2+9z2=22

Find Equations for the Tangent Plane and the Normal Line $x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } = 22$