If α,γ are roots of the equation Ax2−4x+1=0 , and β , δ are roots of the equation of Bx2−6x+1=0 , then the  values of A and B such that α,β,γ and δ are in H.P. are

α,β,γ,δ are in H.P. ⇒1α,1β,1γ,1δ are in A.P. and  they may be taken as a−3d,a−d,a+d,a+3d .  Replacing x by 1/x we get the equation whose roots are 1α,1β,1γ,1δ etc.  Thus, x2−4x+A=0 has roots a−3d,a+d .  And x2−6x+B=0 has roots a−d,a+3d .   Sum =2(a−d)=4,2(a+d)=6∴ a=5/2,d=1/2 Product =(a−3d)(a+d)=A=3(a−d)(a+3d)=B=8