Question

If, in a $\Delta A B C, \tan \frac{A}{2}=\frac{5}{6}$ and $\tan \frac{C}{2}=\frac{2}{5}$, then $a, b, c$ are such that :

• A. $b^{2}=a c$
• B. $2b = a+c$
• C. $2\, ac = b (a+c)$
• D. $a + b = c$

Answer 1

Answer: (B)

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