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 :

  • Option 1)

    0

  • Option 2)

    \alpha \beta

  • Option 3)

    \alpha \gamma

  • Option 4)

    \beta \gamma

 

Answers (1)

\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

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