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, the find the value of arg(z_(1)/z_(4)) + arg(z_(2)//z_(3)) .

If z_(1) , z_(2) , z_(3) are any three complex numbers on Argand plane, then z_(1)(Im(barz_(2)z_(3)))+z_(2)(Imbarz_(3)z_(1)))+z_(3)(Imbarz_(1)z_(2))) is equal to

Let z_(1)z_(2)andz_(3) be three complex numbers and a,b,cin R, such that a+b+c=0andaz_(1)+bz_(2)+cz_(3)=0 then show that z_(1)z_(2)and z_(3) are collinear.

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

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1 z_2|= |z_1||z_2| .

If z_(1),z_(2)inC,z_(1)^(2)+z_(2)^(2)inR,z_(1)(z_(1)^(2)-3z_(2)^(2))=2 and z_(2)(3z_(1)^(2)-z_(2)^(2))=11, the value of z_(1)^(2)+z_(2)^(2) is

For any two complex numbers z_(1) "and" z_(2) , abs(z_(1)-z_(2)) ge {{:(abs(z_(1))-abs(z_(2))),(abs(z_(2))-abs(z_(1))):} and equality holds iff origin z_(1) " and " z_(2) are collinear and z_(1),z_(2) lie on the same side of the origin . If abs(z-(3)/(z))=6 and sum of greatest and least values of abs(z) is 2lambda , then lambda^(2) , is

bb"statement-1" " Let " z_(1),z_(2) " and " z_(3) be htree complex numbers, such that abs(3z_(1)+1)=abs(3z_(2)+1)=abs(3z_(3)+1) " and " 1+z_(1)+z_(2)+z_(3)=0, " then " z_(1),z_(2),z_(3) will represent vertices of an equilateral triangle on the complex plane. bb"statement-2" z_(1),z_(2),z_(3) represent vertices of an triangle, if z_(1)^(2)+z_(2)^(2)+z_(3)^(2)+z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)=0

For any two complex numbers z_(1) "and" z_(2) , abs(z_(1)-z_(2)) ge {{:(abs(z_(1))-abs(z_(2))),(abs(z_(2))-abs(z_(1))):} and equality holds iff origin z_(1) " and " z_(2) are collinear and z_(1),z_(2) lie on the same side of the origin . If abs(z-(1)/(z))=2 and sum of greatest and least values of abs(z) is lambda , then lambda^(2) , is