In a triangle ABC, if $\tan A : \tan B : \tan C = 1 : 2 : 3$ , then $a^{2} : b^{2} : c^{2} =$

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$5 : 8 : 12$

$3 : 5 : 8$

5 : 8 : 10

$5 : 8 : 9$

answer is A.

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Detailed Solution

$\frac{\tan A}{1} = \frac{\tan B}{2} = \frac{\tan C}{3} = k$ (say)

$\in \tan A = π \tan A \Rightarrow 6 k = 6 k^{3}$

$\Rightarrow k = 1$

$∴ \tan A = 1, \tan B = 2, \tan C = 3$

$\Rightarrow c o t A = 1, c o t B = \frac{1}{2}, c o t C = \frac{1}{3},$

$\begin{array}{l} \Rightarrow {cosec}^{2} A = 2, {cosec}^{2} B = \frac{5}{4}, {cosec}^{2} C = \frac{10}{9} \\ \Rightarrow \sin^{2} A = \frac{1}{2}, \sin^{2} B = \frac{4}{5}, \sin^{2} C = \frac{9}{10} \\ ∴ a^{2} : b^{2} : c^{2} = \sin^{2} A : \sin^{2} B : \sin^{2} C \\ = \frac{1}{2} : \frac{4}{5} : \frac{9}{10} = 5 : 8 : 9 . \end{array}$