If `z_(1),z_(2),……………,z_(n)` lie on the circle `|z|=R`, then
`|z_(1)+z_(2)+…………….+z_(n)|-R^(2)|1/z_(1)+1/z_(2)+…….+1/z-(n)|` is equal to

Options

A. nR
B. `-nR`
C. 0
D. n

Answer

C

Explanation

Since, `z_(1),z-(2),z_(3),…………,z_(n)` lie on the circle |z|=R.
`rArr |z_(i)|^(2)=R^(2)`
`rArr z_(i)barz_(i)=R^(2)` for i=1,2,……..n
`rArr 1/z_(i)=barz_(i)/R^(2)` for i=1,2,……….n
Now, `|z_(1)+z_(2)+……….+z_(3)|-R^(2)` for i=1,2,…………..n
Now,
`|z_(1)+z_(2)+.............+z_(2)|-R^(2)|barz_(1)/R^(2)+barz_(2)/R^(2)+...........+barz_(n)/R^(2)|`
`rArr |z_(1)+z_(2)+...............+z_(z)|-|barz_(1)+barz_(2)+............+barz_(n)|`
`rArr |z_(1)+z_(2)+...........+z_(n)|-|bar(z_(1)+z_(2)+.....+z_(n))|`
`=|z_(1)+z_(2)+.............+z_(z)|-|z_(1)+z_(2)+.......+z_(n)|=0`