A) b
B) 2b
C) 3b
D) 4b
Correct Answer: B
Solution :
\[\because \] \[\frac{A}{2}+\frac{B}{2}=\frac{\pi }{2}-\frac{C}{2}\] \[\Rightarrow \] \[\frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}\tan \frac{B}{2}}=\tan \left( \frac{\pi }{2}-\frac{C}{2} \right)\] \[\Rightarrow \] \[=\frac{\frac{5}{6}+\frac{20}{37}}{1-\frac{5}{6}\times \frac{20}{37}}=\cot \frac{C}{2}\] \[\Rightarrow \] \[\frac{305}{122}=\cot \frac{C}{2}\] \[\Rightarrow \] \[\tan \frac{C}{2}=\frac{2}{5}\] Now, \[\tan \frac{A}{2}\tan \frac{C}{2}=\frac{5}{6}\times \frac{2}{5}\] \[\Rightarrow \] \[\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}=\frac{1}{3}\] \[\Rightarrow \] \[\frac{s-b}{s}=\frac{1}{3}\] \[\Rightarrow \] \[3s-3b=s\] \[\Rightarrow \] \[2s=3b\] \[\Rightarrow \] \[a+b+c=3b\] \[\Rightarrow \] \[a+c=2b\]
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