Question

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 :

• A. $0$
• B. $\alpha \gamma$
• C. $\beta \gamma$
• D. $\alpha \beta$

Answer 1

The correct option is C $\beta \gamma$

$\alpha ,\beta ,\gamma$ are in G.P. $\Rightarrow {\beta }^2=\alpha \gamma$
$\alpha x^2+2\beta x+\gamma =0$ and $x^2+x–1=0$
Both the equations have a common roots.
For $x^2+x–1=0$
$\Rightarrow x=\frac{-1\pm \sqrt{5}}{2}$
$\Rightarrow$ Both roots are irrational.
For $\alpha x^2+2\beta x+\gamma =0$
$D=4{\beta }^2-4\alpha \gamma =4{\beta }^2-4{\beta }^2=0$
$\Rightarrow$ Both the roots are rational.
The above conclussions are contradicting with the given statement that the two equations have one common root.
Hence, the question cannot be solved.