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

The asymptotes of a rectangular hyperbola are perpendicular to each other.
Given one asymptote 

$\mathrm{x}+\mathrm{y}+\mathrm{c}=0$

Let the other asymptote be

$\mathrm{x}-\mathrm{y}+\mathrm{\lambda }=0$

We also know that the asymptotes pass through the center of the hyperbola. Therefore, the line $2\mathrm{x}-\mathrm{y}=0$ and the asymptotes
must be concurrent.

Thus, we have 

$\begin{array}{l}\left|\begin{array}{ccc}2& -1& 0\\ 1& 1& \mathrm{c}\\ 1& -1& \mathrm{\lambda }\end{array}\right|=0\\ \mathrm{or} \mathrm{\lambda }=-\frac{\mathrm{c}}{3}\end{array}$