If $\mathbf{a}=\mathbf{i}+\mathbf{j},\mathbf{b}=\mathbf{j}+\mathbf{k},\mathbf{c}=\alpha \mathbf{a}+\beta \mathbf{b}$ and the vectors $\mathbf{i}-2\mathbf{j}+\mathbf{k},3\mathbf{i}+2\mathbf{j}-\mathbf{k}\mathbf{ }\mathbf{,}\mathbf{ }\mathbf{c}$ are coplanar then $\frac{\alpha }{\beta }$ is

$\mathbf{c}=\alpha (\mathbf{i}+\mathbf{j})+\beta (\mathbf{j}+\mathbf{k})=\alpha \mathbf{i}+(\alpha +\beta )\mathbf{j}+\beta \mathbf{k}$

Given vectors are coplanar $\Rightarrow \left|\begin{array}{ccr}\alpha & \alpha +\beta & \beta \\ 1& -2& 1\\ 3& 2& -1\end{array}\right|=0$
$\Rightarrow \alpha (2-2)-(\alpha +\beta )(-4)+\beta \left(8\right)=0\Rightarrow \alpha +\beta +2\beta =0\Rightarrow \alpha +3\beta =0\Rightarrow \frac{\alpha }{\beta }=-3$