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

Since, α, β, γ, δ  are in H.P., 1/α,  1/β,  1/γ,  1/δ  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 1/α,  1/β,  1/γ,  1/δ. Therefore, equation x2−4x+A=0  has roots a−3d, a+d  and equation x2−6x+B=0  has roots a−d, a+3d. Sum of the roots is 2(a−d)=4,  2(a+d)=6 ∴    a=5/2,  d=1/2 Product of the roots is (a−3d)(a+d)=A=3                                      (a−d)(a+3d)=B=8