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  • If alpha , beta are the roots of k(x^2-x)+x+5=0 and if k1, k2 are two values of k obtained from alpha/beta +beta/alpha =4/5 , then k1/(k2)^2+k2/(k1)^2 equals

If alpha , beta are the roots of k(x^2-x)+x+5=0 and if k1, k2 are two values of k obtained from alpha/beta +beta/alpha =4/5 , then k1/(k2)^2+k2/(k1)^2 equals

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Solution:    alpha +eta =k-1/k ,alpha eta =5/k

        Also ,   (4/5)=alpha /eta +eta /alpha =(alpha ^2+eta ^2)/alpha eta =(alpha +eta )^2-2alpha eta /alpha eta

          Rightarrow         4/5=(k-1)^2-10k/5k

            Rightarrow k^2-16k+1=0   

           herefore    k_1+k_2=16,k_1cdot k_2=1

         Thus    ,     k_1/k_2^2+k_2/k_1^2=(k_1)^3+(k_2)^3/(k_2k_1)^2

        Rightarrow        (2^4)^3-3(16)=4048  

       

Posted by

Deependra Verma

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