find the coordinates of the closest point on a cylinder to another point in 3D using lagrange multipliers

Question

The cylinder has the equation $y^2 + z^2 = 25$ The point is $(2,9,12)$ How do I find the coordinates of the closest point on the cylinder to the point $(2,9,12)$ using lagrange multipliers?

Answer 1

Well, the point we seek obviously has $x-$coordinate $x=2.$

The function we want to minimize is

$\displaystyle L = (y-9)^2 + (z-12)^2 - \lambda (y^2 + z^2 -25)$.

$\displaystyle0 = \frac{1}{2} \frac{\partial L}{\partial y} = (y-9) - \lambda y$

$\displaystyle0 = \frac{1}{2} \frac{\partial L}{\partial z} = (z-12) - \lambda z$

$\displaystyle0 = \frac{\partial L}{\partial \lambda} = y^2 + z^2 - 25$.

$\Longrightarrow (1 + \lambda)y = 9 \textrm{ and } (1+\lambda ) z = 12,$

$(81 + 144)/(1+\lambda)^2 = 25$

$\Rightarrow \lambda=2 \Rightarrow y=3, z= 4.$