If n is a natural number `gt 2`, such that `z^(n) = (z+1)^(n)`, then

Options

A. roots of equation lie on a straight line parallel to the y-axis
B. roots of equaiton lie on a straight line parallel to the x-axis
C. sum of the real parts of the roots is `-[(n-1)//2]`
D. none of these

Answer

A, C

Explanation

Given , `z^(n) = (z+ 1)^(n) rArr |z|^(n)= |(z+1)^(n)|`
`therefore |z|^(n) = |z+1|^(n) rArr |z|=|z+1|`
`rArr |z|^(n) = |z+1|^(2)`
`rArr x^(2) +y^(2) = ( x+ 1)^(2) + y^(2)`, where z = x iy
`rArr x = - (1)/(2)`
Hence, z lies on the line `x=- 1//2`. Hence, sum of real parts of the roots is `- (n-1)//2` (since equation has n-1 roots).