alpha and beta be the roots of equation ax2 + 2bx + c=0 and alpha plus omega and beta plus omega be those px2 + qx + r = 0
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Step-by-step explanation:
α,β are roots of ax
2
+bx+c=0
∴α+β=
a
−b
,αβ=
a
c
D
1
=b
2
−4ac=a
2
(
a
2
b
2
−
a
4c
)
=a
2
((α+β)
2
−4αβ)
=a
2
(α−β)
2
γ,δ are roots of px
2
+qx+r=0
γ+δ=
p
−q
,γδ=
p
r
D
2
=q
2
−4pr=p
2
(
p
2
q
2
−
p
4r
)=p
2
((γ+δ)
2
−4γδ)
=p
2
((γ−δ)
2
)
α,β,γ,δ are in A.P.
∴α−β=γ−δ
∴D
1
:D
2
=a
2
:p
2
⇒
D
2
D
1
=
p
2
a
2
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