If a is real constant and A,B,C are variable angle and 

$\sqrt{a^2-4}TanA+aTanB+\sqrt{a^2+4}TanC=6a$
 Then the least value of  $Ta{n}^{2}A+Ta{n}^{2}B+Ta{n}^{2}C=$
 

$\begin{array}{l}\overline{1}=\sqrt{a^2-4}i+aj+\sqrt{a^2+4}k\\ \overline{m}=\mathrm{Tan}Ai+\mathrm{Tan}Bj+\mathrm{Tan}Ck\\ \therefore |\overline{1}\cdot \overline{m}|\le \left|\overline{1}\right|\left|\overline{m}\right|\\ 6a\le \sqrt{a^2-4+a^2+a^2+4}\cdot \sqrt{{\mathrm{Tan}}^2\mathrm{A}+{\mathrm{Tan}}^{2}B+{\mathrm{Tan}}^2\mathrm{C}}\\ {\mathrm{Tan}}^2\mathrm{A}+{\mathrm{Tan}}^2\mathrm{B}+{\mathrm{Tan}}^2\mathrm{C}\ge 36/3=12\\ (\text{ given }\overline{l}\cdot \overline{m}=6a)\end{array}$