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) 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)) .

For any two complex numbers z_1 and z_2 , prove that Re (z_1 z_2) = Re z_1 Re z_2 - Imz_1 Imz_2

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.

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

Complex numbers z_(1),z_(2),z_(3) are the vertices of A,B,C respectively of an equilteral triangle. Show that z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1).

If z_(1),z_(2),z_(3),…………..,z_(n) are n nth roots of unity, then for k=1,2,,………,n

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