An equation of the tangent plane to the surfaceat (1, 1, 1) is

A) $x - y - 2 z = 0$
B) $x + y - 2 z = 0$
C) $x + y + 2 z = 0$
D) $x - y + 2 z = 0$
E) $- x + y - 2 z = 0$

Question

An equation of the tangent plane to the surfaceat (1, 1, 1) is

A) $x - y - 2 z = 0$
B) $x + y - 2 z = 0$
C) $x + y + 2 z = 0$
D) $x - y + 2 z = 0$
E) $- x + y - 2 z = 0$

Quizplus · Accepted Answer

The equation of a tangent plane to a surface $f(x, y, z) = 0$ at a point $(x_0, y_0, z_0)$ can be given by $\nabla f \cdot (x - x_0, y - y_0, z - z_0) = 0$, where $\nabla f$ is the gradient of $f$ at $(x_0, y_0, z_0)$. Without the explicit function $f(x, y, z)$, we cannot directly compute the gradient. However, we can analyze the given options to see which one could represent a plane tangent to a surface at the point (1, 1, 1).Option B, $x + y - 2z = 0$, when evaluated at the point (1, 1, 1), gives $1 + 1 - 2(1) = 0$, which satisfies the condition for a plane passing through the point (1, 1, 1). Without the specific surface equation, we rely on the fact that the correct option must satisfy the condition of passing through the given point and being in the form of a plane equation, which option B does.

An Equation of the Tangent Plane to the Surfaceat (1 x−y−2z=0x - y - 2 z = 0x−y−2z=0

An Equation of the Tangent Plane to the Surfaceat (1 $x - y - 2 z = 0$