$H: 9 x^2-16 y^2+72 x-32 y-16=0$
If equation of hyperbola: $H=0$, then equation of pair of asymptotes: $H+\lambda=0$
Equation of conjugate hyperbola: $H+2 \lambda=0$
$\therefore$ Equation of pair of asymptotes :
$9 x^2-16 y^2+72 x-32 y+(\lambda-16)=0$
If above equation represents pair of straight line then
$\begin{aligned}
& 9(-16)(\lambda-16)-9(-16)^2+16(36)^2=0 \\
\Rightarrow & 9(-16)[(\lambda-16)+16-36 \times 4]=0 \\
\Rightarrow & \lambda=36 \times 4=144 .
\end{aligned}$
$\therefore \quad$ Equation of conjugate hýperbola, $H+2 \lambda=0$
$\begin{aligned}
& \Rightarrow 9 x^2-16 y^2+72 x-32 y-16+2(144)=0 \\
& \Rightarrow 9 x^2-16 y^2+72 x-32 y+272=0 .
\end{aligned}$