Finding the closest point to the origin on the surface Z ∧ 2-xy = 1 Can we use conditional extremum to solve the problem

Finding the closest point to the origin on the surface Z ∧ 2-xy = 1 Can we use conditional extremum to solve the problem

The answer is 1
It is equivalent to having a sphere: x ^ 2 + y ^ 2 + Z ^ 2 = R ^ 2; tangent to Z Λ 2-xy = 1, find the smallest R
By eliminating Z, we get R ^ 2 = x ^ 2 + y ^ 2 + XY + 1;
It is equivalent to finding the minimum value of G = x ^ 2 + y ^ 2 + XY + 1, which is continuously differentiable. Finding the partial derivative leads to a stationary point x = 0, y = 0, which has only one stationary point and takes the minimum value, that is, the minimum value, so instead of x = 0, y = 0, r = 1;
Or (x + Y / 2) ^ 2 + 3 * y ^ 2 / 4 + 1 by formula method
Obviously, when y = 0, x = 0, it is the smallest
It does not involve the domain of definition and constraints, Lagrange multiplier method and conditional extremum