<s>[INST] Surface tangent plane Determine a plane that is tangent to the surface $x ^{2}$ + $3y ^ 2$ + $2z ^ 2$ = $\frac{11}{6}$ and parallel to the plane x + y + z = 10
$$\nabla f (x, y, z) = (2x, 6y, 4z)$$
Quotation vector $X + Y + Z = 10 \implies (1,1,1)$
$$(2x, 6y, 4z) = K (1,1,1)$$
for $K = 1$


*

*$2x = 1 x = \frac{1}{2}$

*$6y = 1 y = \frac{1}{6}$

*$4z = 1 z = \frac{1}{4}$
Putting the plane equation together: $(1,1,1) [(x - 1/2) + (y-1/6) + (z-1/4)]$
My plan equation gave: $X + Y + Z = \frac{-11}{12}$
But the answer of the book is $X + Y + Z = \frac{-11}{6}$or X + Y + Z = $\frac{11}{6}$
 [/INST] I think there is an error for $x$ and $y$, I obtain $K=\pm 2$ and


*

*$x=\pm1$

*$y=\pm \frac13$

*$z=\pm \frac12$
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