Since z=xy+1, thus any point on surface is (x,y,xy+1)

The distance of this point from the origin is,

sqrt(x^2+y^2+(xy+1)^2) = sqrt(x^2+y^2+x^2y^2+2xy+1)

We need to minimize this.

Instead let us minize itssquare, that is to remove the square root.

We need to minimize,

g(x,y)=x^2+y^2+x^2y^2+2xy+1

Find its partial derivatives and make them zero.