The complex numbers `z_1, z_2` and the origin form an equilateral triangle only if (A) `z_1^2+z_2^2-z_1z_2=0` (B) `z_1+z_2=z_1z_2` (C) `z_1^2-z_2^2=z_1z_2` (D) none of these

Explanation

Since , `z_(1) , z_(2) , z_(3)` are the vertices of an equilateral triangle .
`therefore` Circumcentre `(z_(0))` = Centroid `((z_(1) + z_(2) + z_(3))/(3)) " " .... (i)`
Also , for equilateral triangle
`z_(1)^(2) + z_(2)^(2) + z_(3)^(2) = z_(1)z_(2) + z_(2)z_(3) + z_(3)z_(1) " ".... (ii)`
On squaring Eq. (i) we get
`9z_(0)^(2) = z_(1)^(2) + z_(2)^(2) + z_(3)^(2) + 2(z_(1) z_(2) + z_(2)z_(3) + z_(3)z_(1))`
`implies 9z_(0)^(2) = z_(1)^(2) + z_(2)^(2) + z_(3)^(2) + 2(z_(1)^(2) + z_(2)^(2) + z_(3)^(2)) " " `[from Eq. (ii)]
`implies 3z_(0)^(2) = z_(1)^(2) + z_(2)^(2) + z_(3)^(2)`