The asymptotes of a hyperbola are parallel to 2x + 3y = 0 and 3x + 2y = 0. The equation of that hyperbola whose centre is at (1, 2) and passing through (5, 3) is  

Now Equations of asymptotes  parallel to given lines are $2\mathrm{x}+3\mathrm{y}+{\mathrm{k}}_1=0 \mathrm{and} 3\mathrm{x}+2\mathrm{y}+k_2=0$.

we know that asymptotes are passes through centre of hyperbola then ${\mathrm{k}}_1=-8 \mathrm{and} {\mathrm{k}}_2=-7.$

$\therefore Asymptotes equations are 2x+3y-8=0 and 3x+2y-7=0$

Now equation of hyperbola is $(2\mathrm{x}+3\mathrm{y}-8) (3\mathrm{x}+2\mathrm{y}-7)+\mathrm{\lambda }=0$

It passes through (5, 3) then

$$\begin{gathered} \Rightarrow \left(11\right) \left(14\right)+\mathrm{\lambda }=0 \\ \Rightarrow \mathrm{\lambda }=-154 \\ \therefore \mathrm{Hyperbola} \mathrm{is} (2\mathrm{x}+3\mathrm{y}-8) (3\mathrm{x}+2\mathrm{y}-7)-154=0. \end{gathered}$$