Question

In $△ABC$, If $aCo{s}^2\frac{C}{2}+cCo{s}^2\frac{A}{2}=\frac{3b}{2}$, then a,b,c are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P.

Answer 1

Answer: (A)

A.P.

$a{\cos }^2\frac{c}{2}+c{\cos }^2\frac{A}{2}=\frac{3b}{2}$

$a\left(\frac{1+\cos c}{2}\right)+c\left(\frac{1+\cos A}{2}\right)=\frac{3b}{2}$

$=(1+\cos 2A=2{\cos }^{2}A)$

$\Rightarrow \frac{a}{2}+\frac{c}{2}+\frac{a\cos c+c\cos A}{2}=\frac{3b}{2}$

$(b=a\cos c+c\cos A)$

$\Rightarrow \frac{a}{2}+\frac{c}{2}+\frac{b}{2}=\frac{3b}{2}$

$\Rightarrow \frac{a+c}{2}=b$

$a+c=2b$

$\text{Hence,}a,b,c\text{are in}A\cdot P\text{.}$