Question

Find the point(s) on the cone 2=42 52 that are closest to the point (4,6,0).

Answer 1

To find the point(s) on the cone that are closest to the given point, use the method of Lagrange multipliers.

Step-by-step explanation:

To find the point(s) on the cone that are closest to the point (4,6,0), we need to minimize the distance between the given point and any point on the cone. The equation of the cone is x^2 - y^2 = 2z^2. We can use the method of Lagrange multipliers to find the points that minimize the distance.

1. Set up the Lagrange function L(x, y, z, λ) = (x-4)^2 + (y-6)^2 + z^2 + λ(x^2 - y^2 - 2z^2).
2. Take the partial derivatives of L with respect to x, y, z, and λ and set them equal to 0.
3. Solve the resulting system of equations to find the values of x, y, z, and λ.
4. Substitute the values of x, y, and z into the equation of the cone to determine the corresponding values of λ.
5. Plug the values of x, y, z, and λ into the Lagrange function to find the minimum distance.