Suppose vector $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ such that: \begin{align} \lvert \mathbf{a} \rvert &= \mu \lvert \mathbf{b} \rvert \\ \lvert \mathbf{c} \rvert &= \lambda \lvert \mathbf{b} \rvert \\ \lvert \mathbf{c} \rvert ^2 &= \lvert \mathbf{a} \rvert \cdot \lvert \mathbf{b} \rvert - \lvert \mathbf{a}-\mathbf{c} \rvert \cdot \lvert \mathbf{b}-\mathbf{c} \rvert \end{align} If $\lvert \mathbf{a} \rvert$, $\lvert \mathbf{b} \rvert$, $\lvert \mathbf{a} - \mathbf{b} \rvert$ are edges of a triangle which area is 1.
If $\lvert \mathbf{a}-\mathbf{b}\rvert$ take the minimum value, then what the maximum value of $9 \lambda^ 2 - 2 \mu ^2$ are ?