If α,β are the roots of the equation  λ(x2−x)+x+5=0. If λ1 and λ2 are two values of  .λ for which the roots α,β are related by αβ+βα=45find the value of λ1λ2+λ2λ1

The given equation can be written as λx2−(λ−1)x+5=0α,β are the roots of this equation.∴ α+β=λ−1λandαβ=5λbut given αβ+βα=45⇒α2+β2αβ=45⇒(α+β)2−2αβαβ=45⇒(λ−1)2λ2−10λ5λ=45⇒(λ−1)2−10λ5λ=45⇒λ2−12λ+1=4λ⇒λ2−16λ+1=0It is a quadratic in λ, let roots be λ1 and λ2, thenλ1+λ2=16andλ1λ2=1∴ λ1λ2+λ2λ1=λ12+λ22λ1λ2=(λ1+λ2)2−2λ1λ2λ1λ2 =(16)2−2(1)1=254