Let `p` and `q` are complex numbers such that `|p|+|q| lt 1`. If `z_(1)` and `z_(2)` are the roots of the `z^(2)+pz+q=0`, then which one of the following is correct ?

Options

A. `|z_(1)| lt 1` and `|z_(2)| lt 1`
B. `|z_(1)| gt 1` and `|z_(2)| gt 1`
C. If `|z_(1)| lt 1`, then `|z_(2)| gt 1` and vice versa
D. Nothing definite can be said

Answer

A

Explanation

`(a)` `z^(2)+pz+q=0`
`z_(1)+z_(2)=p` and `z_(1)z_(2)=q`
Given `|p|+|q| lt 1`
`:. |z_(1)+z_(2)|+|z_(1)z_(2)| lt 1`……….`(i)`
but `||z_(1)|-|z_(2)|| le |z_(1)+z_(2)|`……..`(ii)`
From `(i)` and `(ii)`
`||z_(1)-z_(2)|| le |z_(1)-z_(2)| lt 1-|z_(1)||z_(2)|` {`1-|z_(1)||z_(2)|` must be positive}
If `|z_(1)|-|z_(2)|-1+|z_(1)||z_(2)| lt 0`
`:.(|z_(1)|-1)+|z_(2)|(|z_(1)|-1) lt 0`
`(|z_(1)|-1)(|z_(2)|+1) lt 0`.........`(iii)`
Also`|z_(2)|-|z_(1)|-1+|z_(1)||z_(2)| lt 0`
`(|z_(1)|-1)(|z_(1)|+1) lt 0` ..........`(iv)`
From `(iii)` and `(iv)`
`|z_(1)) lt 1` and `|z_(2)| lt 1`