# If $\alpha ,\beta \: \: and\: \: \gamma$ are three consecutive terms of a non-constantG.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 : Option 1) 0 Option 2) $\alpha \beta$ Option 3) $\alpha \gamma$ Option 4) $\beta \gamma$

V Vakul

$\alpha ,\beta \: \: and\: \: \gamma$  are in G.P. => $\beta ^{2}=\alpha \gamma$

For equation, $\alpha x^{2}+2\beta x+\gamma =0$

$\Delta =4\beta ^{2}-4\alpha \gamma =0$

Hence, roots are equal & equals to $-\frac{\beta }{\alpha }=-\frac{\gamma }{\beta }$

Since, given equation have common roots , hence $-\frac{\gamma }{\beta }$  must be root of $x^{2}+x-1=0$

$=> \frac{\gamma ^{2}}{\beta ^{2}}-\frac{\gamma }{\beta }-1=0$

$=> \gamma ^{2}-\gamma \beta -\beta ^{2}=0$

$=> \gamma ^{2}=\beta (\gamma +\beta )$

$=> \gamma .\frac{\beta ^{2}}{\alpha }=\beta (\gamma +\beta )$

$=> \gamma .\beta =\alpha (\gamma +\beta )$

Option 1)

0

Option 2)

$\alpha \beta$

Option 3)

$\alpha \gamma$

Option 4)

$\beta \gamma$

Exams
Articles
Questions