# inequality – Minima of \$x+y+frac{1}{xy}\$ given \$x^2+y^2=1\$

If $$x,y$$ are positive reals satisfying $$x^2+y^2=1$$, then the minimum value of $$x+y+frac{1}{xy}$$ is?

#### Attempt 1

I used the Lagrange multipliers method which ended up being cumbersome.

#### Attempt 2

I applied the AM-GM inequality to obtain $$frac{1}{xy}ge2$$ and $$x+ylesqrt2$$ after which I’m lost again.

Im looking for hints/better approaches to the problem.