A diagonal of rhombus ABCD is a member of both the family of lines $(x+y-1)+{\lambda }_1(2x+3y-2)=0$ and  $(x-y+2)+{\lambda }_2(2x-3y+5)$  where ${\lambda }_1,{\lambda }_2\in R$ and one of the vertex of the rhombus is (3,2). If area of the rhombus is  $12\sqrt{5}$ square units then the length of longer diagonal of the rhombus is ___________

Since diagonal is a member of both the families so it will pass through  $(1,0)$ and  $(-1,1)$
Equation of diagonal AC is $x+2y-1=0$
Since one of the vertex  $(3,2)$ which does not be on AC, so equation of BD is $2x-y=4$
Point of intersection of AC and BD is  $P(\frac{9}{5},-\frac{2}{5})$
If vertex B is  $(3,2)$  then vertex D is  $(\frac{3}{5},-\frac{14}{5})$  also  $BD=\frac{12\sqrt{5}}{5}\left(\text{say\hspace{0.17em}}{d}_1\right)$
Area of rhombus $=\frac{1}{2}\times d_1\times d_2$
$$\begin{gathered} \frac{1}{2}\times \left(\frac{12\sqrt{5}}{5}\right)\times d_2=12\sqrt{5} \\ {d}_2=10 \end{gathered}$$