Let `alpha` and `beta` be real and z be a complex number. If `z^(2)+az+beta=0` has two distinct roots on the line Re(z)=1, then it is necessary that

Options

A. `beta sub (0,1)`
B. `beta in (-1,0)`
C. `|beta|-1`
D. `beta in (1,infty)`

Answer

D

Explanation

Let `z_(1)=1+iy` by a rot of `z^(2)+az+beta=0`. Then,
`z_(1)^(2)+az_(1)+beta=0`
`rArr 1+2iy-y^(2)+alpa(1+iy)+beta=0`
`rArr (1-y^(2)+alpha+beta)+i(2y+alphay)=0`
`rArr 1-y^(2)+alpha+beta=0` and `alpha=-2` `[therefore y ne 0]`
`rArr beta=y^(2)+1 rArr beta in (1,infty)`