The asymptotes of a hyperbola are parallel to $3\mathrm{x}+2\mathrm{y}=0, 2\mathrm{x}+3\mathrm{y}=0$ whose centre is at (1,2) and it passes through the point (5,3) then  its equation is

$$\begin{gathered} \mathrm{Equations} \mathrm{of} \mathrm{asymptotes} \mathrm{are} \\ 3\mathrm{x}+2\mathrm{y}+{\mathrm{k}}_1=0 \mathrm{and} 2\mathrm{x}+3\mathrm{y}+{\mathrm{k}}_2=0 \\ \mathrm{since} \mathrm{these} \mathrm{are} \mathrm{possing} \mathrm{through} centre (1, 2) \mathrm{then} \\ \Rightarrow {\mathrm{k}}_1=-7, {\mathrm{k}}_2=-8 \\ \therefore \mathrm{Equation} \mathrm{of} \mathrm{pair} \mathrm{of} \mathrm{asymptotes} \mathrm{is} (3\mathrm{x}+2\mathrm{y}-7) (2\mathrm{x}+3\mathrm{y}-8)=0 \\ \mathrm{Now} \mathrm{equation} \mathrm{of} \mathrm{hyperbola} \mathrm{is} (3\mathrm{x}+2\mathrm{y}-7) (2\mathrm{x}+3\mathrm{y}-8)+\mathrm{k}=0 \\ \mathrm{It} \mathrm{passes} \mathrm{through} (5, 3) \mathrm{then} \\ \Rightarrow \left(14\right) \left(11\right) + \mathrm{k}=0 \Rightarrow \mathrm{k}=-154 \\ \therefore \mathrm{Hyperbola} \mathrm{is} 6{\mathrm{x}}^2+13\mathrm{xy}+6{\mathrm{y}}^2-38\mathrm{x}-37\mathrm{y}-98=0 \end{gathered}$$