If a is a real constant and A, B and C are variable angles and $\sqrt{{\mathrm{a}}^2-4}\tan \mathrm{A}+\mathrm{atan}\mathrm{B}$$+\sqrt{{\mathrm{a}}^2+4}\tan \mathrm{c}=6\mathrm{a}$, then the least value of tan² A + tan² B + tan² C is 

The given relation can be rewritten as the vector expression

$\begin{array}{l}\left(\sqrt{{\mathrm{a}}^2-4}\widehat{\mathrm{i}}+\mathrm{a}\widehat{\mathrm{j}}+\sqrt{{\mathrm{a}}^2+4}\widehat{\mathrm{k}}\right)\cdot (\tan \mathrm{A}\widehat{\mathrm{i}}+\tan \mathrm{B}\widehat{\mathrm{j}}+\tan \mathrm{C}\widehat{\mathrm{k}})=6\mathrm{a}\\ \Rightarrow \sqrt{{\mathrm{a}}^2-4+{\mathrm{a}}^2+{\mathrm{a}}^2+4}\sqrt{{\tan }^2\mathrm{A}+{\tan }^2\mathrm{B}+{\tan }^2\mathrm{C}}\cdot (\cos \mathrm{\theta })=6\mathrm{a}(\because \overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}=|\overrightarrow{\mathrm{a}}\left|\right|\overrightarrow{\mathrm{b}}|\cos \mathrm{\theta })\\ \sqrt{3}\mathrm{a}\sqrt{{\tan }^2\mathrm{A}+{\tan }^2\mathrm{B}+{\tan }^2\mathrm{C}}\cdot (\cos \mathrm{\theta })=6\mathrm{a}\\ {\tan }^2\mathrm{A}+{\tan }^2\mathrm{B}+{\tan }^2\mathrm{C}=12{\sec }^2\mathrm{\theta }\ge 12\left(\because {\sec }^2\mathrm{\theta }\ge 1\right)\end{array}$

The least value of tan² A + tan² B + tan² C is 12.