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

Question

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

Infinity Learn · Accepted Answer

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 $$(a$$−$$3d)(a+d)=A=3$$                                          $$(a$$−$$d)(a+3d)=B=8$$

Given that $α, γ$ are roots of the equation $A x^{2} - 4 x + 1 = 0,$ and $β, δ$ the roots of the equation of $B x^{2} - 6 x + 1 = 0,$ such that $α, β, γ,$ and $δ$ are in H.P., then

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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

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Given that $α, γ$ are roots of the equation $A x^{2} - 4 x + 1 = 0,$ and $β, δ$ the roots of the equation of $B x^{2} - 6 x + 1 = 0,$ such that $α, β, γ,$ and $δ$ are in H.P., then