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

Let $r$ be the common ratio of the given G.P.

$\therefore \beta =\alpha r\text{ and }\gamma =\alpha r^2$                                       …(i)

Now, given $\alpha x^2+2\beta x+\gamma =0$

$\Rightarrow \alpha x^2+2\alpha rx+\alpha r^2=0$                                $\text{ [Using (i)] }$

$\Rightarrow x^2+2rx+r^2=0\Rightarrow (x+r{)}^2=0\Rightarrow x=-r$

it must be root of the equation $x^2+x-1=0$

$\therefore r^2-r-1=0\text{ or }{r}^2=r+1$ 

Now, $\alpha (\beta +\gamma )=\alpha \left(\alpha r+\alpha r^2\right)$

$={\alpha }^2\left(r+r^2\right)$

From option (a), $\beta \gamma =\alpha r\cdot \alpha r^2$

$\begin{array}{l}={\alpha }^2{r}^3={\alpha }^{2}r\cdot r^2={\alpha }^{2}r(r+1)\\ ={\alpha }^2\left(r^2+r\right)\end{array}$