If in a triangle ABC  ,a\cos ^{2}\left ( \frac{C}{2} \right )+c\cos ^{2}\left ( \frac{A}{2} \right )= \frac{3b}{2},  then the sides  a,b \: and\: c

  • Option 1)

    are\: in \: G.P.

  • Option 2)

    are\: in \: H.P.

  • Option 3)

    satisf\! y\: \: a+b=c

  • Option 4)

    are\: in \: A.P.

 

Answers (1)

As we learnt in 

Height and Distances -

The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

-

 

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

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

\Rightarrow a[1+\cos c]+c[1+\cos a]=3b

\Rightarrow a+c+a\cos c+c\cos a =3b

now by sine law , we have  \frac{a}{\sin a}=\frac{b}{\sin b}=\frac{c}{\sin c}

Thus , using it , we have

\sin a+\sin c+\sin a\cos c+\sin c\cos a=3\sin b

\Rightarrow \sin a+\sin c+\sin (a+c)=3\sin b

\Rightarrow \sin a+\sin c+\sin (\pi-b)=3\sin b

\Rightarrow \sin a+\sin c+\sin b=3\sin b

\Rightarrow \sin a+\sin c=2\sin b

using sine law, we have

a+c=2b

a,b,c\ are\ in\ A.P

 

 


Option 1)

are\: in \: G.P.

This is incorrect option

Option 2)

are\: in \: H.P.

This is incorrect option

Option 3)

satisf\! y\: \: a+b=c

This is incorrect option

Option 4)

are\: in \: A.P.

This is correct option

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