If in a $$\Delta ABC$$ $$a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2},$$ then the sides $$a, b$$ and $$c$$ :
JEE · Math · previous-year question
- A.satisfy $$a+b=c$$
- B.are in A.Pcorrect
- C.are in G.P
- D.are in H.P
Answer
B. are in A.P
Explanation
If $$a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2}$$ $$a\left[ {\cos C + 1} \right] + c\left[ {\cos A + 1} \right] = 3b$$ $$\left( {a + c} \right) + \left( {a\cos C + c\cos \,B} \right) = 3b$$ $$a + c + b = 3b$$ or $$a + c = 2b$$ or $$a,b,c$$ are in $$A.P.$$
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