If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$, use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 124$ at the point $( 10,2 )$.
Select the correct answer.
A) $100 x + 36 y = 124$
B) $100 x - 124 y = 12$
C) $10 x + 12 y = 124$
D) $10 x + 12 y = 12$
E) $100 x - 12 y = 12$

Question

Quizplus · Accepted Answer

The gradient of f at (10,2) is $
abla f(10,2) = \langle 20, 24 angle$. This vector is normal to the level curve of f at (10,2), since it points in the direction of greatest increase of f at that point. Thus, the tangent line to the level curve at (10,2) should also be normal to $\langle 20, 24 angle$, which has direction $\langle 24, -20 angle$. Using the point-normal form of the equation of a plane, we have $24(x-10) - 20(y-2) = 0$ which simplifies to $10x+12y=124$. Therefore, the answer is C.

If f(x,y)=x2+6y2f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }f(x,y)=x2+6y2

If $f ( x , y ) = x ^ { 2 } + 6 y ^ { 2 }$