Let `z_1 and z_2` be the root of the equation `z^2+pz+q=0` where the coefficient p and q may be complex numbers. Let A and B represent `z_1 and z_2` in the complex plane. If `/_AOB=alpha!=0 and 0 and OA=OB, where O` is the origin prove that `p^2=4qcos^2 (alpha/2)`

Explanation

Let `z_(1)` and `z_(2)` be roots of the equation `z^(2)+ pz + q =0` .
Then,
`z_(1) +z_(2) = -p,z_(1)z_(2) =q`
Also, `(z_(2))/(z_(1)) = e^(itheta) or z_(2)= z_(1)e^(itheta) `
`rArr z_(1) (1+ e^(itheta)) = -p,z_(1)^(2) e^(itheta) = q`
`z_(1)^(2) = q e^(itheta) = (p^(2))/((1+e^(itheta))^(2))`
`rArr p^(2) = qe^(-itheta) (1+e^(2itheta) + 2e^(itheta))`
`= q(e^(-itheta) + e^(itheta) + 2)`
`=q(2cos theta + 2)`
`= 4q cos ^(2) .(theta)/(2)`