Find symmetric equations of the normal line to the hyperboloid of one sheet defined by the equation $x^{2}+y^{2}-z^{2}=24$ at the point $(3,-4,1)$.

Solution The surface is given by $F(x,y,z) =x^{2}+y^{2}-z^{2}-24=0.$ From Example 1, ${\boldsymbol \nabla } F( 3,-4,1) =6\mathbf{i}-8\mathbf{j}-2\mathbf{k}$. Since the normal line is in the direction of the gradient, symmetric equations of the normal line to the hyperboloid at $(3,-4,1)$ are $$\frac{x-3}{6}=\frac{y+4}{-8}=\frac{z-1}{-2}$$