$$\begin{gathered} \mathrm{The} \mathrm{centre} \mathrm{of} \mathrm{a} \mathrm{rectangular} \mathrm{hyperbola} \mathrm{lies} \mathrm{on} \mathrm{the} \mathrm{line} \mathrm{y}=2\mathrm{x}. \mathrm{If} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{asymptotes} \mathrm{is} \mathrm{x}+\mathrm{y}+\mathrm{c}=0, \mathrm{then} \mathrm{the} \\ \mathrm{other} \mathrm{asymptote} \mathrm{is} \end{gathered}$$

Given one asymptote is x+y+c=0
The asymptotes of a rectangular hyperbola are perpendicular to each other.
So, let the other asymptote be x−y+λ=0$\to \left(1\right)$
We also know that the asymptotes are  passing  through centre of the hyperbola. 

Therefore, the line 2x−y=0 and the asymptotes must be concurrent.
$$\begin{gathered} \mathrm{Solving} \mathrm{y}=2\mathrm{x},\mathrm{x}+\mathrm{y}+\mathrm{c}=0 we \mathrm{get} \mathrm{centre} of hyperbola \\ \Rightarrow x+2x+c=0 \\ \Rightarrow x=\frac{-c}{3} \\ if x=\frac{-c}{3} then y=\frac{-2c}{3} \\ from \left(1\right),\frac{-c}{3}+\frac{2c}{3}+\lambda =0 \\ \Rightarrow \lambda =\frac{-c}{3} \\ \therefore Required asymptote \mathrm{is} \mathrm{x}-\mathrm{y}-\frac{\mathrm{c}}{3}=0 \end{gathered}$$