Let ` z_1 , z _2 and z_3 ` be three complex numbers such that `z_1 + z_2+ z_3 = z_1z_2 + z_2z_3 + z_1 z_3 = z_1 z_2z_3 = 1`. Then the area of triangle formed by points `A(z_1 ), B(z_2) and C(z_3)` in complex plane is _______.

Answer

1

Explanation

`z_(1)+z_(2)+z_(3) =z_(1)z_(2) +z_(3)z_(1) =z_(1)z_(2)z_(3) = 1`
i.e., Sum of roots = Sum of roots taken two at a time = Product of roots = 1
So, equation will be
`z^(3) -z^(2) + z-1=0`
`rArr (z-1) (z^(2) + 1)=0`
`rArrz = 1, -i,i`
Area of the trinagel formed by the point `A(z_(1)),B(z_(2))`
and `C_(z_(3)) = (1)/(2) xx 2xx 1 = 1`