z4+az3+bz2+cz+d=0 are
complex numbers
lying on the circle ∣z∣=1 in the complex plane. The sum of the reciprocals of the roots is necessarily:
options
a) a
b) b
c) -c
d) d
lying on the circle ∣z∣=1 in the complex plane. The sum of the reciprocals of the roots is necessarily:
options
a) a
b) b
c) -c
d) d
Solution
Let α β,γ,δ be the roots of
given Equation
Now all Roots are complex and lie on ∣z∣=1
and as coefficients are real so complex Roots are occur in pair
so Let α=x1+iy1 β=x1−iy1 and α.β=x1^2+y1^2=1
Similarly γ=x2+iy2 δ=x2−iy2 ] and γ.δ=x2^2+y2^2=1
Now all Roots are complex and lie on ∣z∣=1
and as coefficients are real so complex Roots are occur in pair
so Let α=x1+iy1 β=x1−iy1 and α.β=x1^2+y1^2=1
Similarly γ=x2+iy2 δ=x2−iy2 ] and γ.δ=x2^2+y2^2=1
Hence αβγδ = 1
Now α β,γ,δ are roots of f(z) = z^44+az^3+bz^22+cz+1 = 0
So 1/α,1/β,1/γ,1/δ are roots of f(1/z)
= 1/ z^44+a/z^3+b/z^22+c/z+1 =0
Or = z^44+cz^3+bz^22+az+1 = 0
So sum of 1/α,1/β,1/γ,1/δ is –c (-ve coefficient of z^3)
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