Question

The tangent plane to the surface $x^2 + y^2 + z = 9$ at the point $(1, 2, 4)$ is:

• A. $2x + 4y + z = 14$
• B. $4x + 2y + z = 12$
• C. $x + 4y + 2z = 17$
• D. $4x + y + 2z = 14$

Hint:

When solving tangent plane problems, always compute the gradient vector of the given surface equation, as it forms the normal vector to the tangent plane.

Answer 1

The Correct Option is A

Step 1: Recall the equation of the tangent plane. 
For a surface defined by $F(x, y, z) = 0$, the equation of the tangent plane at a point $(x_0, y_0, z_0)$ is: $$F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0,$$ where $F_x, F_y, F_z$ are the partial derivatives of $F(x, y, z)$. 

Step 2: Define the surface and compute partial derivatives. 
The surface is given by: $$F(x, y, z) = x^2 + y^2 + z - 9.$$ Compute the partial derivatives: $$F_x = \frac{\partial F}{\partial x} = 2x, \quad F_y = \frac{\partial F}{\partial y} = 2y, \quad F_z = \frac{\partial F}{\partial z} = 1.$$ 

Step 3: Evaluate the partial derivatives at $(1, 2, 4)$. 
At $(1, 2, 4)$: $$F_x(1, 2, 4) = 2(1) = 2, \quad F_y(1, 2, 4) = 2(2) = 4, \quad F_z(1, 2, 4) = 1.$$ 

Step 4: Write the equation of the tangent plane. 
Substitute the values into the tangent plane equation: $$2(x - 1) + 4(y - 2) + 1(z - 4) = 0.$$ Simplify: $$2x - 2 + 4y - 8 + z - 4 = 0 \quad \Rightarrow \quad 2x + 4y + z = 14.$$ 

Conclusion: The equation of the tangent plane is $2x + 4y + z = 14$.