Question

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

Answer 1

Since A, B, C are A.P
⇒ 2B = A + C
Since A + B + C = $\mathrm{\pi }$ (By angle sum property)
⇒ 3B = $\mathrm{\pi }$
$\mathrm{i}.\mathrm{e}B=\frac{\mathrm{\pi }}{3}...\left(1\right)$
also,
Since a, b, c are in g.p
⇒ b² = ac ....(2)
Using cosB = $\frac{a^2+c^2-b^2}{2ac}$

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

$\Rightarrow \frac{1}{2}=\frac{a^2+c^2-ac}{2ac}$
⇒ ac = a² + c² - ac
⇒ a² + c² = 2ac
⇒ a² + c² = 2b² from (2)
Hence $\lambda$ = 2