Let S be the set of all λ ,   μ for which the vectors λ i ^ - j ^ + k ^ ,   j ^ + 2…

The given equations are coplanar.

$\Rightarrow \left|\begin{array}{ccc}\lambda & -1& 1\\ 1& 2& \mu \\ 3& -4& 5\end{array}\right|=0$

$\Rightarrow \lambda \left[10+4\mu \right]+1\left[5-3\mu \right]+1\left[-10\right]=0$

$\Rightarrow 4\lambda \mu +10\lambda -3\mu =5 .......\left(\mathrm{i}\right)$

And $\lambda -\mu =5$

So, by $\left(1\right)$

$\Rightarrow 4\left(5+\mu \right)\mu +10\left(5+\mu \right)-3\mu =5$

$\Rightarrow 4{\mu }^2+27\mu +45=0$

$\Rightarrow 4{\mu }^2+15\mu +12\mu +45=0$

$\therefore \mu =-3, \frac{-15}{4}$

$\Rightarrow \lambda =2, \frac{5}{4}$

$\therefore \sum _{\lambda \mu \in S}80\left({\lambda }^2+{\mu }^2\right)=80\left[\left(9+4\right)+\frac{225}{16}+\frac{25}{16}\right]$

$=80\left[13+\frac{125}{8}\right]=10\times 229$

$=2290$

Hence this is the correct option.