check the independence of transformed variable of two independent Gamma random variables

Let $X$ and $Y$ are two independent random variables following Gamma Distribution $X\sim \Gamma(\alpha,0,1)$ and $Y\sim \Gamma(\beta,0,1)$

Show that the independence of $U=\frac{X}{X+Y}$ and $V=X+Y$ also implies the independence of $\frac{X^2+Y^2}{XY}$ and $X+Y$.

I can check the independence of $U=\frac{X}{X+Y}$ and $V=X+Y$ when $X$ and $Y$ are two independent random variables by showing that

Joint distribution of $U$ and $V$ is equal to product of marginal distribution of $U$ and $V$.

But i don't know how to check the independence of $\frac{X^2+Y^2}{XY}$ and $X+Y$ when $U=\frac{X}{X+Y}$ and $V=X+Y$ are independent.

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