The ellipsoid $8 x ^ { 2 } + 3 y ^ { 2 } + z ^ { 2 } = 14$ intersects the plane $y = 2$ in an ellipse. Find parametric equations for the tangent line to this ellipse at the point $( 1,2,2 )$.
A) $x = 1 + t , y = 2 , z = - 4 t + 2$
B) $x = 1 + t , y = 2 , z = - 4 t - 2$
C) $x = 1 + t , y = 1 , z = 4 t + 2$
D) $x = t , y = 2 , z = - 4 t + 2$
E) $x = 1 + t , y = 2 , z = 4 t - 2$

Question

Quizplus · Accepted Answer

The gradient of the ellipsoid at the point (1,2,2) is perpendicular to the tangent plane. The gradient is $(16x, 6y, 2z)$, which at (1,2,2) is (16,12,4). The direction vector of the tangent line is perpendicular to this gradient, and the correct parametric form is given by choice A.

The Ellipsoid 8x2+3y2+z2=148 x ^ { 2 } + 3 y ^ { 2 } + z ^ { 2 } = 148x2+3y2+z2=14

The Ellipsoid $8 x ^ { 2 } + 3 y ^ { 2 } + z ^ { 2 } = 14$