Partial Derivatives & the Gradient · Hard problem

A firm's cost (in dollars) from using $x$ units of input $X$ and $y$ units of input $Y$ is $C(x,y)=x^{2}+2y^{2}$, subject to the budget line $x+2y=20$. At the point $(8,6)$ on this budget line, moving in the direction $(-2,1)/\sqrt5$ decreases the budget-$X$ input and increases $Y$ (staying on the line). Find the rate (a signed number, per unit distance moved) at which cost changes in that direction.