Question

In a $\Delta ABC,tan\frac{A}{2}=\frac{5}{6}andtan\frac{C}{2}=\frac{2}{5},$ then a,b,c are such that

• A. b²=ac
• B. 2b=a+c
• C. 2ac=b (a+c)
• D. a+b=c

Answer 1

The correct option is B

2b=a+c

$Since,tan\frac{A}{2}=\frac{5}{6}andtan\frac{C}{2}=\frac{2}{5}\therefore tan\frac{A}{2}tan\frac{C}{2}=\frac{5}{6}.\frac{2}{5}\Rightarrow \frac{\Delta }{s(s-a)}.\frac{\Delta }{s(s-c)}=\frac{1}{3}\Rightarrow \frac{s-b}{s}=\frac{1}{3}\Rightarrow \frac{s-b}{s}=\frac{1}{3}\Rightarrow 3s-3b=s\Rightarrow 2s=3b\Rightarrow a+c=2b$