Sum of three variables given two equations Given $$x^2+y^2+z^2=121$$ 
$$x\sqrt{11} + 4y + z\sqrt{22}=77$$
Find $$ \frac{\sqrt{11} + 4 + \sqrt{22}}{x+y+z} $$  
I tried to plug in something for z at first, since x and y should have unique values for every value of z, but that didn't seem to work.
The answer is 7/11, which is clearly the second equation divided by the first but I don't understand how or why that would lead the final expression.

$x^2+y^2+z^2=121$
Is the equation of a sphere centered at the origin of radius 11.
$x\sqrt{11} + 4y + z\sqrt{22}=77$ is the equation of a plane
Origin is  $\frac {77}{\sqrt {11 + 4^2 +22}} = 11$ units from the plane!
The plane is tangent at to the sphere.
the point of tangency $(x,y,z) = \frac {11}{7}\cdot(\sqrt{11} , 4, \sqrt {22})$
$x+y+z = \frac {11}{7} (\sqrt {11} + 4 + \sqrt{22})$
$\frac {\sqrt {11} + 4 + \sqrt{22}}{x+y+z} = \frac {7}{11}$
