Given α1,β1,γ1,δ1 are the roots of equation ax4+bx3+cx2+dx+e=0 and α2,β2,γ2,δ2 are the roots of equation ex4+dx3+cx2+bx+a=0 ∴ Clearly, α1,β1,γ1,δ1 are the reciprocal of roots of ax4+bx3+cx2+dx+e=0 Also, δ1>γ1>β1>α1>0 and δ2>γ2>β2>α2>0 ∴δ1 and α2 are reciprocal Similarly γ1 and β2, γ2 and β1 and α1 and δ2 are reciprocal
∴α1δ2=1,β1γ2=1,β2γ1=1,α2δ1=1
α1−δ2=2
∴α1−
1
α1
=2⇒α12−2α1−1=0
and δ1−α2=4 δ1−
1
δ1
=4=δ12−4δ1−1=0 ∴α1 and δ1 are roots of ax4+bx3+cx2+dx+e=0