The angles A, B, C of a ∆ABC are in AP and the sides a, b,
Question:

The angles ABC of a ∆ABC are in AP and the sides abc are in G.P. If a2 + c2 = λb2, then λ = ____________.

Solution:

Since ABC are A.P

$\Rightarrow 2 B=A+C$

Since $A+B+C=\pi$  (By angle sum property)

$\Rightarrow 3 B=\pi$

i. e $B=\frac{\pi}{3}$     ….(1)

Also,

Since abc are in g.p 

$\Rightarrow b^{2}=a c$   …(2)

Using $\cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$

i.e $\cos \frac{\pi}{3}=\frac{a^{2}+c^{2}-a c}{2 a c} \quad$ from (1) and (2).

$\Rightarrow \frac{1}{2}=\frac{a^{2}+c^{2}-a c}{2 a c}$

$\Rightarrow a c=a^{2}+c^{2}-a c$

$\Rightarrow a^{2}+c^{2}=2 a c$

$\Rightarrow a^{2}+c^{2}=2 b^{2}$   from (2)

Hence $\lambda=2$

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