Example 7: Tangent Plane of a Ellipsoid

Find the tangent plane to the ellipsoid

x² + 2y² + 3z² = 6

at the point (1, 1, 1) (see Figure 11.6.10).

Put

F(x, y, z) = x² + 2y² + 3z² - 6.

Then

F_x(x, y, z) = 2x, F_y(x, y, z) = Δy, F_z(x, y, z) = 6z.

F_x(1, 1,1) = 2,          F_y(1, 1,1) = 4,          F_z(1l, 1,1) = 6.

The tangent plane has the equation

2(x - 1) + 4(y - 1) + 6(z - 1) = 0.

Figure 11.6.10

Find ∂z/∂x and ∂z/∂y at (1,1,1).

At (1,1,1),