Using Lagrange multipliers, the points on the surface xy - z^2 = 1 closest to the origin are (1, -1, 0) and (1, 1, 0) corresponding to lambda = 2 and lambda = -2, respectively.
To find the points on the surface xy - z^2 = 1 that are closest to the origin, we can use the method of Lagrange multipliers. The problem is to minimize the distance from the origin, which is given by the square of the distance function D^2 = x^2 + y^2 + z^2, subject to the constraint F(x, y, z) = xy - z^2 - 1 = 0.
The Lagrangian function is given by:
L(x, y, z, λ) = x^2 + y^2 + z^2 + λ(xy - z^2 - 1)
The critical points occur when the partial derivatives of L with respect to x, y, z, and λ are all zero:
∂L/∂x = 2x + λy = 0
∂L/∂y = 2y + λx = 0
∂L/∂z = 2z - 2λz = 0
∂L/∂λ = xy - z^2 - 1 = 0
From the first two equations, we get x = -λy/2 and y = -λx/2. Substituting these into the third equation, we get z = 0.
Now, substituting z = 0 and the expressions for x and y into the fourth equation, we get:
(-λx/2)y - 0^2 - 1 = 0
Solving this, we get xy = 2, which implies λ = ±2.
Now, since λ = ±2, let's consider λ = 2. Substitute λ = 2 into the expressions for x and y:
x = -2y/2 ⟶ x = -y
Now, substitute x = -y and λ = 2 into the constraint equation xy - z^2 = 1:
-y^2 - 0^2 = 1
The closest point on the surface to the origin is where y = -1 and z = 0, which implies x = 1.
Therefore, one solution is (x, y, z) = (1, -1, 0). However, we need to check the other case (λ = -2) to find the correct answer.
Substitute λ = -2 into the expressions for x and y:
x = -(-2y)/2 ⟶ x = y
Now, substitute x = y and λ = -2 into the constraint equation xy - z^2 = 1:
y^2 - 0^2 = 1
The closest point on the surface to the origin is where y = 1 and z = 0, which implies x = 1.
Therefore, the correct answer is (x, y, z) = (1, 1, 0).