Solve the problem.
-Write parametric equations for the tangent line to the curve of intersection of the surfaces $x + y ^ { 2 } + 7 z = 9$ and $x = 1$ at the point $( 1,1,1 )$.
A) x = 1, y = 7t + 1, z = -2t + 1
B) x = 1, y = 14t + 1, z = -t + 1
C) x = 1, y = 7t + 1, z = -t + 1
D) x = 1, y = 14t + 1, z = -2t + 1

Question

Quizplus · Accepted Answer

To find the tangent line, we first need the gradient vectors of both surfaces at the point (1,1,1). For the surface $x + y^2 + 7z = 9$, the gradient is $\nabla f = (1, 2y, 7)$ and at (1,1,1) it becomes (1,2,7). For the surface $x=1$, the gradient is always in the direction of the x-axis, i.e., (1,0,0). The tangent line to the curve of intersection is in the direction of the cross product of these two gradients. The cross product of (1,2,7) and (1,0,0) gives a direction vector perpendicular to both gradients, which is (0, -7, 2). Thus, the parametric equations of the tangent line are $x = 1$, $y = -7t + 1$, and $z = 2t + 1$. However, since the direction can be reversed, the correct answer is $x = 1$, $y = 7t + 1$, $z = -2t + 1$, which matches option A.

Solve the Problem x+y2+7z=9x + y ^ { 2 } + 7 z = 9x+y2+7z=9

Solve the Problem $x + y ^ { 2 } + 7 z = 9$