If `z=x+iy (x, y in R, x !=-1/2)`, the number of values of z satisfying `|z|^n=z^2|z|^(n-2)+z |z|^(n-2)+1.` `(n in N, n>1)` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

B

Explanation

The given equation is
`\|z|^(n)=(z^(2)+z)^(n-2+1)`
`rArr z^(2)+z` is real
`rArr z^(2)+z=barz^(2)+barz`
`rArr (z-z)(z+barz+1)=0`
`rArr z=barz=x " as " z+barz+1ne0(xne-1//2)`
Hence, the given equation reduces to `x^(n)=x^(n)+x|x|^(n-2+1)`
`rArr x|x|^(n-2)=-1`
`rArr x=-1`
So number of solution is 1.