A) 5 : 9 : 13
B) 5 : 6 : 7
C) 4 : 5 : 6
D) 3 : 4 : 5
Correct Answer: C
Solution :
\[a<b<c\]are in A.P. \[\angle C=2\angle A\](Given) \[\Rightarrow \sin C=\sin 2A\] \[\Rightarrow \sin C=2\sin A.\cos A\] \[\Rightarrow \frac{\sin C}{\sin A}=2\cos A\]\[\Rightarrow \frac{c}{a}=2\frac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}\] put\[a=b-\lambda ,c=b+\lambda ,\lambda >0\] \[\Rightarrow \]\[\lambda =\frac{b}{5}\] \[\Rightarrow \]\[a=b-\frac{b}{5}=\frac{4}{5}b,c=b+\frac{b}{5}=\frac{6b}{5}\] \[\Rightarrow \]required ratio = 4 : 5 : 6
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