Find the maximum and minimum distances from the orgin to the ellipsoid x^(2)+y^(2)/9+z^(2)/4=1

Question

If I could get a worked out step by step solution for this it would really help to see it worked out.

Answer 1

Hi Mandy,

 

You're trying to optimize the distance f(x,y,z) = d² = x² + y² + z² and constrained by the points on the ellipsoid g(x,y,z) = x² + y²/9 + z²/4 = 1.

 

To find the critical points, we use Lagrange multipliers to find where ∇f = λ∇g. The partial derivatives of each function are all fairly simple, and give the equations:

 

∂x: x = x λ

∂y: y = y/9 λ

∂z: z = z/4 λ

 

For the first equation to be true, either x = 0 or λ = 1.

For the second equation to be true, either y = 0 or λ = 9.

For the third equation to be true, either z = 0 or λ = 4.

 

The three values of λ give different possible solutions.

 

If λ=1, then y=z=0, and this gives the points on the ellipsoid

x² + 0²/9 + 0²/4 = 1

and x = ±1.

These two possible solutions are (1,0,0) and (-1,0,0).

 

Similarly, when λ=9, you get (0,3,0) and (0,-3,0). When λ=4 you get (0,0,2) and (0,0,-2).

 

You can then test these to find which ones give the maximum and minimum distances.