Suppose $U =\begin{pmatrix}X\\Y\end{pmatrix}$ is a random gaussian vector. The question is : find the conditions of existence of a real $\alpha$ such that $X-\alpha Y$ and $X$ are two independent random variables.
What I do:
We must have :
$$Cov(X-\alpha Y, X)=0$$ $$Cov(X,X) - \alpha \, Cov(Y,X)=0$$ $$\alpha = \frac{Var(X)}{Cov(X,Y)}$$ My conclusion: there exists an $\alpha$ if and only if $X$ and $Y$ are not independent.
Is it the good answer?