Solve the problem.
-Write parametric equations for the tangent line to the curve of intersection of the surfaces $z = 10 x ^ { 2 } + 2 y ^ { 2 }$ and $z = x + y + 10$ at the point $( 1,1,12 )$.
A) x = -19t + 1, y = 21t + 1, z = 16t + 12
B) x = -3t + 1, y = 21t + 1, z = 16t + 12
C) x = -19t + 1, y = 19t + 1, z = 16t + 12
D) x = -3t + 1, y = 19t + 1, z = 16t + 12

Question

Quizplus · Accepted Answer

To find the parametric equations for the tangent line, we first need to find the gradient vectors of both surfaces at the point $(1,1,12)$. For the surface $z = 10x^2 + 2y^2$, the gradient vector is $\nabla z_1 = (20x, 4y, -1)$ and at the point $(1,1,12)$, it becomes $(20, 4, -1)$. For the surface $z = x + y + 10$, the gradient vector is $\nabla z_2 = (1, 1, -1)$. The tangent line to the curve of intersection is in the direction of the cross product of these two gradient vectors. The cross product of $(20, 4, -1)$ and $(1, 1, -1)$ is $(-3, 19, 16)$. Thus, the parametric equations for the tangent line are $x = -3t + 1$, $y = 19t + 1$, and $z = 16t + 12$.

Solve the Problem z=10x2+2y2z = 10 x ^ { 2 } + 2 y ^ { 2 }z=10x2+2y2

Solve the Problem $z = 10 x ^ { 2 } + 2 y ^ { 2 }$