z^n=(z+1)^n. roots of the equation lie on the line

a) 2x+1=0

b)2x-1=0

c)x+1=0

d)x-1=0

The answer is A

let w=(z+1)/z , then wⁿ=1 and w not equal to 1

w = [ cos(2rπ)+i sin(2rπ) ]^1/n

    = cos(2rπ/n)+i sin(2rπ/n)    ( de moivres theorem )

w  = cos(2rπ/n)+i sin(2rπ/n)    where r=1,2,3,.......n-1

(z+1)/z  = cos(2rπ/n)+i sin(2rπ/n)

1/z  = -1+ cos(2rπ/n)+i sin(2rπ/n)

       = -2 sin²(rπ/n) + 2i sin(rπ/n).cos(rπ/n)

       =  2i sin(rπ/n)[ cos(rπ/n)+i sin(rπ/n)]

=>   x +iy = z = 1 / {2i sin(rπ/n).[ cos(rπ/n)+i sin(rπ/n)]}

                       =  [ cos(rπ/n)+i sin(rπ/n)]^-1 /  [2i sin(rπ/n)]

                       =  [ cos(rπ/n) -i sin(rπ/n)] / [2i sin(rπ/n)]               ( De moivres theorem )

                       = -1/2  - i.cot(rπ/n)/2  where   r = 1,2,3,.......n-1

this shows that all roots lie on 2x+1=0

Any solution to the equation zⁿ = (z+1)ⁿ satisfies |z| = |z+1| i.e. the point is equidistant from (0,0) and (-1,0).

This means it lies on the perpendicular bisector of the line joining (0,0) and (-1,0) i.e. x = -1/2 or 2x+1 = 0.