Sum and difference of perfectly correlated variables

Suppose $X$ and $Y$ are perfectly correlated, $\rho(X,Y) = 1$ , with $\sigma_Y > \sigma_X > 0$ . What is the correlation $\rho(X+Y,\ X-Y)$ ?

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$\rho=1$ means $Y=aX+b$ with $a=\sigma_Y/\sigma_X>1$ . Write $X+Y$ and $X-Y$ in terms of $X$ .
Two affine functions of the same variable have correlation $\pm1$ ; the sign is $\operatorname{sign}(1-a^2)$ . Or check $\operatorname{Cov}=\sigma_X^2-\sigma_Y^2$ .

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