Having a bit of trouble with min/max distance from sphere to point

Question

The sphere is

$x^2 + y^2 + z^2 = 81$

and the point is $(5,6,9)$

I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon.

I found the gradient of $F: (2(x-5), 2 (y-6), 2(z-9))$
Gradient of $G: (2x, 2y, 2z)$

$2x-10 = 2 \lambda x$
$2y-12 = 2 \lambda y$
$2z-18 = 2 \lambda z$
$x^2+y^2+z^2 = 81$

This system seems very challenging to solve.

Answer 1

If you draw the line through the center of the sphere (which is the origin) and $(5,6,9)$ it intersects the sphere in two points. One of these is the min distance, the other is the max distance. It doesn't even matter if the point is inside or outside the sphere. So find the distance from the origin to $(5,6,9)$ and add/subtract the radius of the sphere. If you draw the corresponding 2D figure you can see what is happening.

Answer 2

The system of equations is rather easy to solve. The first three equations give:

$$x = \frac{5}{1-\lambda}$$ $$y = \frac{6}{1-\lambda}$$ $$z = \frac{9}{1-\lambda}$$

Now just plug these values into the fourth equation to get $\lambda$. (But first you have to decide whether $x^2+y^2+z^2$ equals $81$ or $18$ $-$ I expect that $18$ is the typo.)