Find Shortest Distance from z2+3x-xy=9 to Origin

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SUMMARY

The shortest distance from the surface defined by the equation z² + 3x - xy = 9 to the origin can be determined using Lagrange multipliers. The gradient of the function, calculated as ∇f(x,y,z) = (3-y, -x, 2z), indicates the direction of steepest ascent but does not directly provide the distance. The correct approach involves applying the method of Lagrange multipliers to find the minimum distance constraint to the origin.

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Homework Statement


a) what is the shortest distance from the surface z2+3x-xy=9 to the origin?

Homework Equations


I know that when you take the gradient of an equation the gradient is perp. to the vector and gives the direction of largest rate of inc.


The Attempt at a Solution


so I took the gradient of f(x,y,z) and i got
fx=3-y
fy=-x
fz=2z
I tried looking at these and thinking of it as a parameterized line and i got an answer of (3,0,0) but I don't think that it is right..
now I'm still a little confused on what i should do next, can someone help point me in the right direction?
 
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The gradient won't help you find distance to the origin. I would recommend Lagrange multipliers here.
 

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