Q70 / 160 · TS EAMCET 2006 · Mathematics

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 :

  • A
    $b^2=a c$
  • Correct answer
    B
    $2 b=a+c$
  • C
    $2 a c=b(a+c)$
  • D
    $a+b=c$

Solution

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

Hence, the correct answer is option: 'B'.

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