Determine the tangent plane for the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 121 \text { at } ( 11,0,0 )$
A) $x = 11$
B) $x = 1$
C) $x = 121$
D) $x = 22$
E) $x = 0$

Question

Quizplus · Accepted Answer

The equation of the tangent plane to the surface defined by $F(x, y, z) = 0$ at a point $(x_0, y_0, z_0)$ can be found using the gradient of $F$, $\nabla F = \left( \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right)$, and the formula $\nabla F \cdot (x - x_0, y - y_0, z - z_0) = 0$. For the given hyperboloid $x^2 + y^2 - z^2 = 121$, the gradient is $\nabla F = (2x, 2y, -2z)$. At the point $(11, 0, 0)$, this becomes $(22, 0, 0)$. Thus, the equation of the tangent plane is $22(x - 11) + 0(y - 0) + 0(z - 0) = 0$, which simplifies to $x = 11$.

Determine the Tangent Plane for the Hyperboloid x2+y2−z2=121 at (11,0,0)x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 121 \text { at } ( 11,0,0 )x2+y2−z2=121 at (11,0,0)

Determine the Tangent Plane for the Hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 121 \text { at } ( 11,0,0 )$