Solve the problem.
-$\text { Find the equation for the tangent plane to the surface } x ^ { 2 } - 10 x y z + y ^ { 2 } = 12 z ^ { 2 } \text { at the point } ( - 1 , - 1 \text {, } - 1 ) \text {. }$
A) x + y + z = 10
B) -12x - 12y + 14z = 10
C) -12x - 12y + 14z = 1
D) x + y + z = 1

Question

Quizplus · Accepted Answer

First, we find the gradient of the given surface at the point $( - 1 , - 1 , - 1 )$.

\begin{align*}

abla f ( x , y , z ) &= \left[ \frac { \partial f } { \partial x } , \frac { \partial f } { \partial y } , \frac { \partial f } { \partial z } ight] \
&= \left[ 2x - 10yz, 2y - 10xz, 24z - 10xy ight] \
\end{align*}

Therefore, at $( - 1 , - 1 , - 1 )$, the gradient is $
abla f (-1, -1, -1) = [12, 12, -34]$.

Now, we can use the point-normal form of the equation of a plane to find the equation of the tangent plane.

\begin{align*}
	ext{Plane:} \hspace{5mm} &  -12(x-(-1)) - 12(y-(-1)) + 14(z-(-1)) = 0 \
\Rightarrow \hspace{5mm} & -12x - 12y + 14z = 10
\end{align*}

Therefore, the correct answer is $\boxed{	ext{B}}$.

Solve the Problem A) X + Y + Z = 10 B) -12x

Solve the Problem A) X + Y + Z = 10
B) -12x