The distance between an arbitrary point on the surface and the origin is

Recall that for differentiable functions

and

, the composition

attains extrema at the same points that

does, so we can consider an augmented distance function

The Lagrangian would then be

We have partial derivatives

Set each partial derivative to 0 and solve the system to find the critical points.
From the second equation it follows that either

or

. In the first case we arrive at a contradiction (I'll leave establishing that to you). If

, then we have

This means

so that the points on the surface closest to the origin are

.