If `alpha, beta` are the roots fo the equation `lamda(x^(2)-x)+x+5=0`. If `lamda_(1)` and `lamda_(2)` are two values of `lamda` for which the roots `alpha, beta` are related by `(alpha)/(beta)+(beta)/(alpha)=4/5` find the value of `(lamda_(1))/(lamda_(2))+(lamda_(2))/(lamda_(1))`

Explanation

The given equation can be written as
`lamda x^(2)-(lamda-1)x+5=0`
`:'alpha, beta` are the roots of the this equation.
`:.alpha+beta=(lamda-1)/(lamda)` and `alpha beta=5/(lamda)`
But given `(alpha)/(beta)+(beta)/(alpha)=4/5`
`implies(alpha^(2)+beta^(2))/(alpha beta)=4/5`
`implies((alpha+beta)^(2)-2alpha beta)/(alpha beta)=4/5 implies(((lamda-1)^(2))/(lamda^(2))-10/(lamda))/(5/(lamda))=4/5`
`=((lamda-1)^(2)-10lamda)/(5 lamda)=4/5implieslamda^(2)-12lamda+1=4lamda`
`implieslamda^(2)-16lamda+1=0`
It is a quadratic in `lamda` let roots be `lamda_(1)` and `lamda_(2)`, then
`lamda_(1)+lamda_(2)=16` and `lamda_(1)lamda_(2)=1`
`:.(lamda_(1))/(lamda_(2))+(lamda_(2))/(lamda_(1))=(lamda_(1)^(2)+lamda_(2)^(2))/(lamda_(1)lamda_(2))=((lamda_(1)+lamda_(2))^(2)-2lamda_(1)lamda_(2))/(lamda_(1)lamda_(2))`
`=((16)^(2)-2(1))/1=254`