Question

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

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

Answer 1

Answer: (B)

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