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 \( 2x + 4y + z = 14 \).