Examples

Tangent plane to a hyperboloid

Tangent plane to a hyperboloid

Problem: Find the tangent plane at $(2,1,3)$ to the one-sheeted hyperboloid given by $4x^2+9y^2-z^2=16$.

Solution: Consider the function $f(x,y,z)=4x^2+9y^2-z^2$, so the hyperboloid is the level curve $f(x,y,z)=16$. The normal vector is then given by the gradient $\nabla f=(8x,18y,-2z)$. At $(2,1,3)$ this takes the value $(16,18,-6)$. The tangent plane is perpendicular to this vector and passes through $(2,1,3)$:

$$\nabla f\cdot ((x,y,z)-(2,1,3))=0,16(x-2)+18(y-1)-6(z-3)=0.$$