Question

Let `A` and `B` be real and `z` be complex number. If `z^2+Az+B=0` has two distinct roots on the line `Re(z)=1`, then find the the interval of `B` which is necessary to belong ?

Answer 1

`B in (1, oo)`

Explanation:

Given that the coefficients of `z^2+Az+B = 0` are real, the roots must occur in complex conjugate pairs.

So if the roots satisfy `Re(z) = 1`, then they are `1+ki` and `1-ki` for any `k > 0`.

So:

`z^2+Az+B = (z-(1+ki))(z-(1-ki))`

`color(white)(z^2+Az+B) = ((z-1)-ki)((z-1)+ki)`

`color(white)(z^2+Az+B) = (z-1)^2+k^2`

`color(white)(z^2+Az+B) = z^2-2z+(k^2+1)`

So:

`A = -2` and `B=k^2+1 > 1`

So `B in (1, oo)`