If $f ( x , y ) = x ^ { 2 } + 9 y ^ { 2 }$, use the gradient vector $\nabla f ( 10,2 )$ to find the tangent line to the level curve $f ( x , y ) = 136$ at the point $( 10,2 )$. Select the correct answer.
A) $100 x + 81 y = 136$
B) $100 x - 136 y = 12$
C) $10 x + 18 y = 136$
D) $100 x - 18 y = 12$
E) $10 x + 18 y = 12$

Question

Quizplus · Accepted Answer

The gradient vector of $f$ at $(10,2)$ is $\nabla f(10,2) = \langle 20, 36\rangle$. This vector is normal to the level curve at $(10,2)$. Therefore, the tangent line to the level curve at $(10,2)$ must be perpendicular to $\nabla f(10,2)$ and has slope $-\frac{20}{36} = -\frac{5}{9}$. Since the point $(10,2)$ lies on the level curve, we have $$f(10,2) = 10^2 + 9(2^2) = 136.$$ Using point-slope form, the equation of the tangent line is $$y-2 = -\frac{5}{9}(x-10).$$ Simplifying, we get $$\boxed{10x+18y=136}.$$

If f(x,y)=x2+9y2f ( x , y ) = x ^ { 2 } + 9 y ^ { 2 }f(x,y)=x2+9y2

If $f ( x , y ) = x ^ { 2 } + 9 y ^ { 2 }$