If 'a' is a real constant (|a| \(\geq \)2) and A, B and C are variable angles and \(\sqrt{a^2 - 4} \tan {A} + a \tan {B} + \sqrt{a^2 - 4} \tan {C} = 6 a\), what is the least value of \(\tan^2{A} + \tan^2{B} + \tan^2{C}\) ?

12

\(\sqrt{a^2 - 4} \tan {A} + a \tan {B} + \sqrt{a^2 - 4} \tan {C} = 6 a\)

\([\sqrt{a^2 - 4}\hat{i} + a\hat{j} + \sqrt{a^2 - 4}\hat{k}][\tan{A}\hat{i} + \tan{B}\hat{j} + \tan{C}\hat{k}] = 6 a\)

\(\sqrt{(a^2 - 4) + a^2 + (a^2 + 4)} \sqrt{\tan^2{A} + \tan^2{B} + \tan^2{C}} \cos{\theta} = 6a\)

where \(\theta\) is the angle between two vectors.

\(\sqrt{3}a \sqrt{\tan^2{A} + \tan^2{B} + \tan^2{C}} = 6a \sec \theta\)

\((3a^2)(\tan^2 {A} + \tan^2{B} + \tan^2{C} = 36 a^2 \sec^2 {\theta}\)

\(\tan^2 {A} + \tan^2{B} + \tan^2{C} = 12 \sec^2 {\theta}\geq 12\)