Let `a`, `b`, `c` be distinct complex numbers with `|a|=|b|=|c|=1` and `z_(1)`, `z_(2)` be the roots of the equation `az^(2)+bz+c=0` with `|z_(1)|=1`. Let `P` and `Q` represent the complex numbers `z_(1)` and `z_(2)` in the Argand plane with `/_POQ=theta`, `o^(@) lt 180^(@)` (where `O` being the origin).Then

Options

A. `b^(2) = ac`
B. `PQ = sqrt(3)`
C. `theta = (pi)/(3)`
D. `theta = (2pi)/(3)`

Answer

A, B, D

Explanation

It is given that `z_(1)` and `z_(2)` are the roots of the equation `az^(2) + bz + c = 0`
So, `|z_(1) + z_(2)|=|-(b)/(a)| =1`
And `|z_(1)z_(2)|=|(c)/(a)|=1`
`therefore |z_(2)| = 1`
`|z_(1)+z_(2)|^(2) = 1`
`therefore 2 + z_(1)barz_(2) + z_(1)barz_(2)=1`
Now, `z_(2) =z_(1)e^(itheta)`
`therefore |z_(1)z_(1)e^(itheta)|=|z_(1)||1+e^(itheta)|=1`
`therefore 2 cos.(theta)/(2)=1`
`therefore = (2pi)/(3)`
Now,`((z_(1) + z_(2)))/(z_(1)z_(2)) = 1`
` rArr (b^(2))/(a^(2)) = (c)/(a)`
`rArr b^(2) = ac`
` P = |z_(1) - z_(2)|= sqrt(3)`