Unit Normals to a Curve

Example

Find all the unit normals to $x^3+xy+y^3=11$ at $(x,y)=(1,2)$

Soln

$\nabla f=(3x^2+y,x+3y^2)$ $\nabla f(1,2)=(5,13)$ This is perpendicular to the curve at $(1,2)$ Then, the unit normal is $\frac{5,13}{\sqrt{194}}$