The correct options are
A 2α2β
B 2αβ2
D 2αβ
Roots of x2+x+1=0 are ω,ω2 where ω is the cube root of unity.
Let α=ω, β=ω2
Now,
1a+α+1b+α+1c+α=2β
⇒1a+ω+1b+ω+1c+ω=2ω [∵ω3=1]
1a+β+1b+β+1c+β=2α⇒1a+ω2+1b+ω2+1c+ω2=2ω2
So, the equation
1a+x+1b+x+1c+x=2x ⋯(1)
has roots ω,ω2
Simplifying the equation, we get
⇒x[(x+a)(x+c)+(x+a)(x+b)+(x+b)(x+c)] =2(x+a)(x+b)(x+c)⇒x[3x2+2(a+b+c)x+⋯]=2[x3+(a+b+c)x2+⋯]⇒x3+0⋅x2+…=0
The roots of the cubic equations are ω,ω2,γ, so
Sum of roots
ω+ω2+γ=0⇒γ=1 (∵ω+ω2=−1)
So, x=1 is also a root of equation (1), putting it in equation (1), we get
1a+1+1b+1+1c+1=2
Now,
αβ=ω3=1α2β=1αβ2=1
Hence,
1a+1+1b+1+1c+1=2αβ=2α2β=2αβ2