# Prove independence of quadratic and linear form

Suppose $$(X,Y)$$ is bi-variate normal distribution with correlation $$\rho$$ and mean $$(0,0)^T$$ and both variances 1, and $$(X_1,Y_1), (X_2,Y_2),...,(X_n,Y_n)$$ is a i.i.d sample from the bi-variate normal distribution. Are $$\sum_{i=1}^n (X_i-\bar{X})^2$$ and $$\bar{Y}$$ independent? $$\bar{X}$$ and $$\bar{Y}$$ are the sample means of $$X$$ and $$Y$$.

My attempt: Let $$x=(X_1,...X_n)^T$$ and $$y=(Y_1,...,Y_n)^T$$, trying to prove $$x^TAx$$ and $$By$$ independent where $$A={\bf I} -1/n {\bf 1} {\bf 1}^T$$ and $$B=1/n {\bf 1}^T$$. My intuition is they are not independent unless $$\rho=0$$. Am I wrong?