If from a point P representing the complex number `z_1` on the curve `|z|=2`, two tangents are drawn from P to the curve `|z|=1`, meeting at points `Q(z_2) and R(z_3)`, then :

Options

A. complex number `(z_(1) + z_(2)+z_(3))//3` will be on the curve `|z|=1`
B. `((4)/(barz_(1)) + (1)/(barz_(2)) + (1)/(barz_(3)))((4)/(z_(1))+(1)/(z_(2))+(1)/(z_(3)))= 9`
C. `arg((z_(2))/(z_(3)))= (2pi)/(3)`
D. orth ocenre and circumcenter of `DeltaPQR` wil coincide

Answer

A, B, C, D

Explanation

Since OQ = 1 and OP = 2, so sin `(/_OPQ) = 1//2` and hence
`/_ QPR = pi//3`. Then `/_ POR` is equilateral . Also, `OM botQR`. Then
from `DeltaOMQ, OM = 1//2` Hence, `MN=1//2`. Then centroid of `DeltaPQR` lies on `|z|=1`
As PQR is an equilateral triagle, so orthocenter, circumcenter, and centroid will coincide. Now
`|(z_(1) + z_(2)+z_(3))/(3)|=1`
or `|z_(1) + z_(2) + z_(3)|^(2) =9`
or `(z_(1) + z_(2) + z_(3)) (barz_(1) + barz_(2) + barz_(3)) = 9`
or `((4)/(barz_(1)) + (1)/(barz_(2)) + (1)/(barz_(3))) ((4)/(z_(1)) +(1)/(z_(2)) + (1)/(z_(3))) =9`
and `/_QOR = 120^(@)`