how to find the shortest distance from the origin to the surface y^2-z^2=10 ?

Question

turingtest · Answer

d to the surface is given by $$d=\sqrt{x^2+y^2+z^2}$$if we minimize the distance we minimize its square, so we only need the absolute minimum of$$d(x,y,z)=x^2+y^2+z^2$$using substitution we can see that the surface allows us to eliminate one of the variables$$d(x,y,z)=x^2+y^2+(y^2-z^2)$$

turingtest · Answer

I intend $d(x,y,z)$ to represent the (square of the) distance function from the origin in case I didn't make that clear

turingtest · Answer

oops, typo
also meant that$$d(,y,z)=x^2+y^2+(y^2-10)$$

anonymous · Answer

shouldn't is be eliminate the x ?

turingtest · Answer

why
I typed above wrong, I meant$$d(x,y)=x^2+y^2+(y^2-10)$$subbing for z
I don't see how to sub for x in this problem