Weighting two correlated assets for minimum variance

Let $X$ and $Y$ be jointly normal with standard deviations $\sigma_X,\sigma_Y$ and correlation $\rho$. Find non-negative weights $a$ and $b$ with $a+b=1$ that make $\operatorname{Var}(aX+bY)$ as small as possible.

$a^*=\dfrac{\sigma_Y^2-\rho\sigma_X\sigma_Y}{\sigma_X^2+\sigma_Y^2-2\rho\sigma_X\sigma_Y}$, $b^*=1-a^*$, clipped to $[0,1]$. Set the derivative of $a^2\sigma_X^2+(1-a)^2\sigma_Y^2+2a(1-a)\rho\sigma_X\sigma_Y$ to zero; the variance is a convex quadratic in $a$. If the formula falls outside $[0,1]$ (e.g. $\rho>\sigma_X/\sigma_Y$ with $\sigma_X<\sigma_Y$), put all weight on the lower-variance variable. Normality is not needed.