Find the equation of the tangent plane to $x ^ { 2 } + 2 x y + 4 y + 6 = z ^ { 2 }$ at the point (-4, 1, 3).
A) $6 x + 4 y + 6 z = - 4$
B) $- 6 x + 4 y + 6 z = - 2$
C) $6 x - 4 y + 6 z = - 2$
D) $6 x + 4 y + 6 z = 2$
E) $6 x + 4 y + 6 z = - 2$

Question

Quizplus · Accepted Answer

The gradient of the function $F(x, y, z) = x^2 + 2xy + 4y + 6 - z^2$ is $(2x + 2y, 2x + 4, -2z)$. At the point (-4, 1, 3), the gradient is $(-6, 4, -6)$. The equation of the tangent plane is $-6(x + 4) + 4(y - 1) - 6(z - 3) = 0$, which simplifies to $6x + 4y + 6z = -2$.

Find the Equation of the Tangent Plane To x2+2xy+4y+6=z2x ^ { 2 } + 2 x y + 4 y + 6 = z ^ { 2 }x2+2xy+4y+6=z2

Find the Equation of the Tangent Plane To $x ^ { 2 } + 2 x y + 4 y + 6 = z ^ { 2 }$