A) a, c, b are in AP
B) a, b, c are in AP
C) b, a, c are in AP
D) a, b, c are in GP
Correct Answer: B
Solution :
\[\tan \frac{A}{2}=\frac{5}{6},\tan \frac{C}{2}=\frac{2}{5}\] \[\Rightarrow \] \[\tan \frac{A}{2}\tan \frac{C}{2}=\frac{1}{3}\] \[\Rightarrow \] \[\sqrt{\frac{(s-b)(s-c)}{s(s-a)}\frac{(s-b)(s-a)}{s(s-c)}}=\frac{1}{3}\] \[\Rightarrow \] \[\frac{s-b}{s}=\frac{1}{3}\]\[\Rightarrow \] \[3s-3b=s\] \[\Rightarrow \] \[2s=3b\]\[\Rightarrow \] \[2b=a+c\]\[\Rightarrow \] \[a,b,c\]are in AP.
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