Find an equation of the tangent plane to the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4$ at the point $( 1,0,1 )$ .
A) $3 x - y + z = 4$

B) $2 x - 4 y + z = 3$
C) $x + 2 y + 3 z = 4$
D) $2 x + y - z = 1$
E) $3 x + 2 y - z = 2$
F) $4 x - y + 2 z = 6$

G) $x + y - z = 0$

H) $5 x + 3 y - 2 z = 3$

Question

Find an equation of the tangent plane to the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4$ at the point $( 1,0,1 )$ .
A) $3 x - y + z = 4$

B) $2 x - 4 y + z = 3$ 
C) $x + 2 y + 3 z = 4$ 
D) $2 x + y - z = 1$ 
E) $3 x + 2 y - z = 2$ 
F) $4 x - y + 2 z = 6$

G) $x + y - z = 0$

H) $5 x + 3 y - 2 z = 3$

Quizplus · Accepted Answer

To find the equation of the tangent plane to the given hyperboloid at the point $(1,0,1)$, we first need to find the gradient of the function defining the hyperboloid, which is $f(x, y, z) = x^2 + y^2 - z^2 - 2xy + 4xz - 4$. The gradient vector $\nabla f$ at any point $(x, y, z)$ is given by $\left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$. Calculating the partial derivatives, we get:- $\frac{\partial f}{\partial x} = 2x - 2y + 4z$,- $\frac{\partial f}{\partial y} = 2y - 2x$,- $\frac{\partial f}{\partial z} = -2z + 4x$.Substituting the point $(1,0,1)$ into these derivatives, we get the gradient vector at $(1,0,1)$ as $(2(1) - 2(0) + 4(1), 2(0) - 2(1), -2(1) + 4(1)) = (6, -2, 2)$.The equation of the tangent plane at $(1,0,1)$ can be written as $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$, where $(a, b, c)$ is the gradient vector at the point and $(x_0, y_0, z_0)$ is the point of tangency. Substituting the values, we get $6(x - 1) - 2(y - 0) + 2(z - 1) = 0$, which simplifies to $6x - 2y + 2z = 6 + 2 - 6 = 4$, or $3x - y + z = 4$, matching option A.

Find an Equation of the Tangent Plane to the Hyperboloid x2+y2−z2−2xy+4xz=4x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4x2+y2−z2−2xy+4xz=4

Find an Equation of the Tangent Plane to the Hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } - 2 x y + 4 x z = 4$