$$\begin{gathered} \mathrm{The} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{line} \mathrm{passing} \mathrm{through} \mathrm{the} \mathrm{centre} \mathrm{of} \mathrm{rectangular} \mathrm{hyperbola} \mathrm{is} \mathrm{x}-\mathrm{y}-1=0. \mathrm{If} \mathrm{one} \mathrm{of} \mathrm{its} \\ \mathrm{asymptote} \mathrm{is} 3\mathrm{x}-4\mathrm{y}-6=0, \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{other} \mathrm{asymptote} \mathrm{is} \end{gathered}$$

$$\begin{gathered} Given asymptote 3x-4y-6=0\to \left(1\right) \\ Since asymptotes are perpendicular to each other then equation \\ of other asymptote is 4x+3y+k=0\to \left(2\right) \\ We also know that the asymptotes are passes through centre of the hyperbola. \\ Therefore, the line x-y-1=0\to \left(3\right) and the asymptotes must be concurrent. \\ Now point of intersection of \left(1\right) and \left(3\right) is (-2,-3) \\ from \left(2\right) ,4(-2)+3(-3)+k=0 \\ \Rightarrow k=17 \\ So,required asymptote is 4x+3y+17=0 \end{gathered}$$