If `z_(1),z_(2)andz_(3)` are the vertices of an equilasteral triangle with `z_(0)` as its circumcentre , then changing origin to `z^(0)` ,show that `z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=0,` where`z_(1),z_(2),z_(3),` are new complex numbers of the vertices.

Options

A. `z_(0)^(2)`
B. `3z_(0)^(2)`
C. `2z_(0)^(2)`
D. 0

Answer

B

Explanation

In an equilateral triangle circumcenter, centroid and orthocenter are coincident.
`therefore z_(0) = (z_(1)+z_(2)+z_(3))(3) rArr z_(1)+z_(2)+z_(3) = 3z_(0)`
It is given that `z_(1),z_(2),z_(3)` are the affixes of the vertices of an equilateral triangle.
`therefore z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)` ...............(i)
`rArr (z_(1)+z_(2)+z_(3))^(2) =3(z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1))`
`rArr z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=3z_(0)^(2)`
`rArr z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=3z_(0)^(2)`
`rArr z_(1)^(2)+z_(2)^(2)+z_(3)^)(2)=3z_(0)^(2)` [Using (i)]