Use lagrange multiplier to find greatest and least distance from point (2,1,-4) to ellipsoid x^2 +5y^2 +z^2 =1.
I tried solving it but i got a polynomial to power 4.
Its too long to post my working so ill just dropbox it.
dropboxcom/s/nobvp0739uk4yff/Photo%203-23-17%2C%2008%2004%2004.jpg?dl=0
dropboxcom/s/0xs6pyvht3nmrxo/Photo%203-23-17%2C%2008%2003%2052.jpg?dl=0
I worked it out, and also got a 4th-degree equation in x. So what? Just solve it for x using whatever method works best.
I did get z=x rather than z = -2x, but maybe I made a mistake.
Turns out my lecturer tells everyone theres no 5y^2... welp i just wasted few hrs...
I see... thanks for the help
To find the greatest and least distance from a given point to an ellipsoid, we can use the method of Lagrange multipliers. Here is how you can approach the problem.
Step 1: Define the distance function
Let's denote the given point as P(2, 1, -4). The distance function between P and any point on the ellipsoid can be defined as:
D(x, y, z) = (x - 2)^2 + (y - 1)^2 + (z + 4)^2
Step 2: Define the constraint function
The equation of the ellipsoid is x^2 + 5y^2 + z^2 = 1. We can define the constraint function as:
g(x, y, z) = x^2 + 5y^2 + z^2 - 1
Step 3: Define the Lagrangian function
The Lagrangian function combines the distance function and the constraint function, along with the Lagrange multiplier (λ):
L(x, y, z, λ) = D(x, y, z) - λ * g(x, y, z)
Step 4: Find the partial derivatives
Take the partial derivatives of L with respect to x, y, z, and λ, respectively:
∂L/∂x = 2(x - 2) - 2λx
∂L/∂y = 2(y - 1) - 10λy
∂L/∂z = 2(z + 4) - 2λz
∂L/∂λ = -(x^2 + 5y^2 + z^2 - 1)
Step 5: Set the partial derivatives equal to zero
Set the partial derivatives equal to zero and solve the resulting system of equations simultaneously:
2(x - 2) - 2λx = 0
2(y - 1) - 10λy = 0
2(z + 4) - 2λz = 0
x^2 + 5y^2 + z^2 - 1 = 0
Step 6: Solve the system of equations
Solve the system of equations to find the critical points (points where the partial derivatives are zero) and the values of λ.
Step 7: Substitute the values back into the distance function
Substitute the values of x, y, and z obtained from the previous step back into the distance function D(x, y, z) to find the greatest and least distances.
I understand that you have already attempted to solve the problem using this approach. Unfortunately, I am unable to browse external links or view images directly. However, I hope that the steps provided above will guide you in solving the problem.