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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

MIT 18.02 Multivariable Calculus, Fall 2007
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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??
I think you are correct!

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