Solve the problem.
-$$\text { Find parametric equations for the normal line to the surface } \mathrm { z } = \mathrm { e } ^ { 8 \mathrm { x } ^ { 2 } + 10 \mathrm { y } ^ { 2 } } \text { at the point } ( 0,0,1 ) \text {. }$$
A) x = t, y = t, z = t - 1
B) x = 0, y = 0, z = t + 1
C) x = t, y = t, z = -t - 1
D) x = 0, y = 0, z = t - 1

Question

Quizplus · Accepted Answer

The normal line to the surface at a point is in the direction of the gradient of the surface at that point. The gradient of $z = e^{8x^2 + 10y^2}$ at the point (0,0,1) is $\nabla z = (0,0)$ in the xy-plane, indicating the normal direction is along the z-axis. Thus, the parametric equations keeping x and y constant (0) and allowing z to vary linearly with t are x = 0, y = 0, z = t + 1.

Solve the Problem A) X = T, Y = T, Z = T