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

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SUMMARY

The minimum value of the expression $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ for the constraints $00$ is achieved through optimization techniques involving calculus. The critical points are determined by setting the partial derivatives with respect to both variables equal to zero. The analysis reveals that the minimum occurs at specific values of $x$ and $y$, which can be calculated using numerical methods or symbolic computation tools.

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Find the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ for $0<x<\sqrt{2}$ and $y>0$.
 
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anemone said:
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:
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 $(3-1)^2+(1-3)^2=8$.
 

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