MHB Find Minimum Value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$

[anemone]

Find the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ for $0<x<\sqrt{2}$ and $y>0$.

 

[anemone]

Find the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ for $0<x<\sqrt{2}$ and $y>0$.

Solution suggested by other:

Spoiler

The given function is the square of the distance between a point of the quarter of the circle $x^2+y^2=2$ in the open first quadrant and a point of the half hyperbola $xy=9$ in that quadrant. Then tangents to the curves at $(1,\,1)$ and $(3,\,3)$ separate the curves, and both are perpendicular to $x=y$, so those points are at the minimum distance, and the answer is $(3-1)^2+(1-3)^2=8$.