Show that the triangle whose vertices are `z_(1)z_(2)z_(3)andz_(1)'z_(2)'z_(3)'` are directly similar , if `|{:(z_(1),z'_(1),1),(z_(2),z'_(2),1),(z_(3),z'_(3),1):}|=0`

Find the coordinates of the centroid of the triangle whose vertices are (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2))and(x_(3),y_(3),z_(3)) .

If |z_(1)|=2,|z_(2)|=3,|z_(3)|=4and|z_(1)+z_(2)+z_(3)|=5. then |4z_(2)z_(3)+9z_(3)z_(1)+16z_(1)z_(2)| is

If |z_(1)|=|z_(2)|andamp(z_(1))+amp(z_(2))=0, then

|z_(1)+z_(2)|=|z_(1)|+|z_(2)| is possible, if

Find the circumcentre of the triangle whose vertices are given by the complex numbers z_(1),z_(2) and z_(3) .

If z^(1) =2 -I, z_(2)=1+i , find |(z_(1) + z_(2) + 1)/(z_(1)-z_(2) + 1)|

Let z_(1),z_(2) and z_(3) be three non-zero complex numbers and z_(1) ne z_(2) . If |{:(abs(z_(1)),abs(z_(2)),abs(z_(3))),(abs(z_(2)),abs(z_(3)),abs(z_(1))),(abs(z_(3)),abs(z_(1)),abs(z_(2))):}|=0 , prove that (i) z_(1),z_(2),z_(3) lie on a circle with the centre at origin. (ii) arg(z_(3)/z_(2))=arg((z_(3)-z_(1))/(z_(2)-z_(1)))^(2) .

If z_(1), z_(2) and z_(3), z_(4) are two pairs of conjugate complex numbers, then find arg ((z_(1))/(z_(4)))+arg((z_(2))/(z_(3))) .

If the complex numbers z_(1) and z_(2) arg (z_(1))- arg(z_(2))=0 then showt aht |z_(1)-z_(2)|=|z_(1)|-|z_(2)| .