Question

Lagrange multiplier question, please help I continously get the imaginary # 5iradical 5 as olambda
surface is f(x,y,z) = x^2+2y^2-z^2+4xy=250
compute coordinartes of every pt on surface at which the tangent plane is parallel to the plane
x-2y+z=20

Answer 1

Hi!
You haven't said how you approached to solve the problem. The way I did is like this:
\[\nabla f = <(2x+4y),(4x+4y),(-2z)>\]
and
\[\nabla g = <1,-2,1>\]
The equations are:
\[\nabla f = \lambda \nabla g, g = c\]
This gave me a the following matrix equation:
\[\left[\begin{matrix}2 & 4 & 0 & -1 \\ 4 & 4 & 0 & 2\\ 0 & 0 & -2 & -1\\ 1 & -2 & 1 & 0\end{matrix}\right] \left[\begin{matrix}x \\ y \\z\\ \lambda \end{matrix}\right] = \left[ \begin{matrix}0 \\ 0 \\0\\ 20 \end{matrix}\right]\]
The solution to this is: x = 7.5, y = -5, z = 2.5 and lambda = -5

Next, I would find the equation to the tangent plane at (x,y,z) = (7.5,-5,2.5) and that should give the final solution...

I'm really not sure if I am correct... I haven't verified my results! Will post if/when I verify my results. Perhaps someone else can let me know if I am right or wrong??

Answer 2

I think you are correct!