Question

In a $△ABC$, $\tan A.\tan B.\tan C=9$. For such triangles, if ${\tan }^{2}A+{\tan }^{2}B+{\tan }^{2}C=k$ then:

• A. $9.\sqrt[3]{33}<k<27$
• B. $\nleqq 27$
• C. $k<9.\sqrt[3]{33}$
• D. $k\le 27$

Answer 1

The correct option is A $9.\sqrt[3]{33}<k<27$

In a triangle,

$\tan A+\tan B+\tan C=\tan A\tan B\tan C$

By Cauchy's inequality;

$(t_1+t_2+t_3)\le \left(\sqrt{t_1^2+t_2^2+t_3^2}\right)\left(\sqrt{3}\right)⟹k<27$

by $GM\le AM⟹{9.3}^{\frac{1}{3}}\le k$