The symmetric equations of the normal line to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) are

A) $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { 3 }$
B) $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { 2 }$
C) $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { - 2 }$
D) $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { - 3 }$
E) $\frac { x - 1 } { - 2 } = y + 2 = \frac { z - 3 } { - 3 }$

Question

The symmetric equations of the normal line to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) are

A) $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { 3 }$
B) $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { 2 }$
C) $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { - 2 }$
D) $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { - 3 }$
E) $\frac { x - 1 } { - 2 } = y + 2 = \frac { z - 3 } { - 3 }$

Quizplus · Accepted Answer

The gradient vector of the surface at (1, -2, 3) is $\nabla f = (8x, 2y, 4z)$. At (1, -2, 3), this becomes (8, -4, 12). The direction ratios of the normal line are (-2, -1, -3), leading to the symmetric equations $\frac{x-1}{-2} = y+2 = \frac{z-3}{-3}$.

The Symmetric Equations of the Normal Line to the Surface 4x2+y2+2z2=264 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 264x2+y2+2z2=26

The Symmetric Equations of the Normal Line to the Surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$