Finding a Normal Line to a Tangent Plane

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 \[ \frac{x-3}{6}=\frac{y+4}{-8}=\frac{z-1}{-2} \]