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