If, in a $\triangle A B C, \tan \frac{A}{2}=\frac{5}{6}$ and $\tan \frac{C}{2}=\frac{2}{5}$, then $a, b, c$…

If, in a $\triangle A B C, \tan \frac{A}{2}=\frac{5}{6}$ and $\tan \frac{C}{2}=\frac{2}{5}$, then $a, b, c$ are such that :
  1. $b^2=a c$
  2. $2 b=a+c$
  3. $2 a c=b(a+c)$
  4. $a+b=c$

Solution

$\because \quad \tan A=\frac{5}{6}$ and $\tan C=\frac{2}{5}$ $\therefore \quad \tan A \tan C=\frac{5}{6} \cdot \frac{2}{5}$ $\Rightarrow \quad \frac{s-b}{s}=\frac{1}{3}$ $\Rightarrow \quad 3 s-3 b=s$ $\Rightarrow \quad 2 s=3 b$ $\Rightarrow \quad a+c=2 b$

Asked in: AP EAMCET 2006

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