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  4. If α, β and γ are three consecutive terms of a non-constant G.P. such that the equations α x2 + 2β x + γ = 0 and x2 + x - 1 = 0 have a common root, then α(β + γ) is equal to :

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Q. If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root, then $\alpha(\beta + \gamma)$ is equal to :

JEE MainJEE Main 2019Sequences and Series

Solution:

$\alpha x^2 + 2\beta x + \gamma = 0$
$Let \beta \, = \, \alpha t, \gamma \, = \alpha t^2$
$\therefore \, \alpha x^2 + 2tx + t^2 = 0$
$\Rightarrow \, \, x^2 + 2tx + t^2 \, = 0$
$\Rightarrow \, \, (x + t)^2 = 0$
$\Rightarrow \, \, x \, = \, -t$
it must be root of equation $x^2 $+ x - 1 = 0
$\therefore \, \, t^2 - t - 1 = 0 $
Now
$\alpha (\beta + \gamma) = \alpha^2(t + t^2)$
Option 1 $\beta \gamma = \alpha t . \alpha t^2 = \alpha^2 t^3 = a^2 (t^2 + t)$
$ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, $ (from equation 1)